Mercurial > hg > mpdl-xml-content

<?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC">
  <metadata>
    <dcterms:identifier>ECHO:U19ERSE3.xml</dcterms:identifier>
    <dcterms:creator identifier="GND:118963066">Barrow, Isaac</dcterms:creator>
    <dcterms:title xml:lang="la">Lectiones opticae et geometricae</dcterms:title>
    <dcterms:date xsi:type="dcterms:W3CDTF">1674</dcterms:date>
    <dcterms:language xsi:type="dcterms:ISO639-3">lat</dcterms:language>
    <dcterms:rights>CC-BY-SA</dcterms:rights>
    <dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license>
    <dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder>
  </metadata>
  <text xml:lang="la" type="free">
<div xml:id="echoid-div1" type="section" level="1" n="1"><pb file="0001" n="1"/>
<pb file="0002" n="2"/>
<pb file="0003" n="3"/>
<pb file="0004" n="4"/>
<pb file="0005" n="5"/>
<pb file="0006" n="6"/>
<pb file="0007" n="7"/>
<pb file="0008" n="8"/>
</div>
<div xml:id="echoid-div2" type="section" level="1" n="2">
<head xml:id="echoid-head1" xml:space="preserve">Imprimatur,</head>
<p>
  <s xml:id="echoid-s1" xml:space="preserve">_Edmundo Boldero_ <emph style="sc">Procancellario</emph>. <lb/></s>
  <s xml:id="echoid-s2" xml:space="preserve">_Pet. </s>
  <s xml:id="echoid-s3" xml:space="preserve">Gunning_ Præfect. </s>
  <s xml:id="echoid-s4" xml:space="preserve">Coll. </s>
  <s xml:id="echoid-s5" xml:space="preserve">S. </s>
  <s xml:id="echoid-s6" xml:space="preserve">_ſoban_. </s>
  <s xml:id="echoid-s7" xml:space="preserve"><lb/>_fo. </s>
  <s xml:id="echoid-s8" xml:space="preserve">Pearſon_ Mag. </s>
  <s xml:id="echoid-s9" xml:space="preserve">Coll. </s>
  <s xml:id="echoid-s10" xml:space="preserve">S. </s>
  <s xml:id="echoid-s11" xml:space="preserve">_Trin_.</s>
  <s xml:id="echoid-s12" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s13" xml:space="preserve">Martii 22. </s>
  <s xml:id="echoid-s14" xml:space="preserve">166 {8/9}.</s>
  <s xml:id="echoid-s15" xml:space="preserve"/>
</p>
<pb file="0009" n="9"/>
</div>
<div xml:id="echoid-div3" type="section" level="1" n="3">
<head xml:id="echoid-head2" xml:space="preserve">LECTIONES <lb/>_OPTICÆ &amp; GEOMETRICÆ:_ <lb/>In quibus <lb/>PHÆNOMENωN OPTICORUM</head>
<head xml:id="echoid-head3" xml:space="preserve">Genuinæ _Rationes_ inveſtigantur, ac exponuntur: <lb/>ET <lb/>_Generalia_ Curvarum Linearum _Symptomata declarantur_.</head>
<head xml:id="echoid-head4" xml:space="preserve">Auctore <emph style="sc">Isaaco</emph> <emph style="sc">Barrow</emph>,</head>
<head xml:id="echoid-head5" xml:space="preserve">Collegii _<emph style="sc">S S</emph>. Trinitatis_ in Academia _Cantab._ Præfecto, <lb/>Et _<emph style="sc">SOCIETATIS REGIÆ</emph>_ Sodale.</head>
<p>
  <s xml:id="echoid-s16" xml:space="preserve">Oi φύσει λομςιὸι είς πὰντα τὰ παθήματα,ώ \~ες ξπω εξπεῖν, \~οζ\~εῖς φα@-<lb/>νοντομ θϊιε βραδεῖς, αv̀ @ν τδτω πομδδ@ῶσι ηὶ γνμνὰσωντου ηᾶν <lb/>μηδὲν ἄλλο ώφελησῶσιν, ω<unsure/>μως εϊσγε τὸ ο@ξύτεςοι αντοὶ αυτν γίγνεσου <lb/>πάντες δπιδιδόχσιν. </s>
  <s xml:id="echoid-s17" xml:space="preserve">Platode Repub.</s>
  <s xml:id="echoid-s18" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s19" xml:space="preserve">”Αρχεῖ, εῖ τà υεν {οὐ} χεῖρον Ariſt.</s>
  <s xml:id="echoid-s20" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div4" type="section" level="1" n="4">
<head xml:id="echoid-head6" xml:space="preserve">LONDINI,</head>
<head xml:id="echoid-head7" xml:space="preserve">Typis _Guilielmi Godbid_, &amp; proſtant venales apud <lb/>_Robertum Scott_, in vico <emph style="bf">Little-Britain.</emph> 1674.</head>
<pb file="0010" n="10"/>
<handwritten/>
<handwritten/>
<handwritten/>
<pb file="0011" n="11"/>
</div>
<div xml:id="echoid-div5" type="section" level="1" n="5">
<head xml:id="echoid-head8" xml:space="preserve">SPECTATISSIMIS VIRIS <lb/><emph style="sc">Roberto</emph> <emph style="sc">Raworth</emph> &amp; <emph style="sc">Thomæ</emph> <emph style="sc">Buck</emph> <lb/>ARMIGERIS;</head>
<p>
  <s xml:id="echoid-s21" xml:space="preserve">HAS, à <emph style="sc">Venerabili</emph> <emph style="sc">Viro</emph> <lb/>_HENRICO LVCAS_ <lb/>inſtitutæ atque dotatæ, ab <lb/>ipſis verò optimâ fide, ſummâque pru-<lb/>dentiâ adminiſtratæ &amp; </s>
  <s xml:id="echoid-s22" xml:space="preserve">conſtitutæ in <lb/><emph style="sc">Academia</emph> <emph style="sc">Cantabrigiensi</emph>, <lb/><emph style="sc">Professionis</emph> MATHEMATICÆ <lb/>primitias, gratitudinis ac obſervantiæ <lb/>ergò, devovet</s>
</p>
</div>
<div xml:id="echoid-div6" type="section" level="1" n="6">
<head xml:id="echoid-head9" style="it" xml:space="preserve">Iſaac Barrow</head>
<pb file="0012" n="12"/>
<pb file="0013" n="13"/>
</div>
<div xml:id="echoid-div7" type="section" level="1" n="7">
<head xml:id="echoid-head10" xml:space="preserve"><emph style="sc">Epistola</emph> ad LECTOREM.</head>
<p style="it">
  <s xml:id="echoid-s23" xml:space="preserve">_<emph style="sc">Benigne</emph> <emph style="sc">Lector</emph>_, <lb/>MInimè tibi deſtinatum boc quicquid est opellæ, <lb/>ſtatim ipſe, modò digneris inſpicere, multis ab <lb/>indiciis deprebendes; </s>
  <s xml:id="echoid-s24" xml:space="preserve">nec tamen ut juris id <lb/>tui fieret, defuerunt auctores. </s>
  <s xml:id="echoid-s25" xml:space="preserve">quibus tandem, <lb/>animo certè trepidans atque renitens, idcircò <lb/>præſertim obſequutus ſum, quoniam in boc, quod ipſe primus <lb/>obiêrim, munus ſucc@ſſuris exemplo præire rem literariam; <lb/></s>
  <s xml:id="echoid-s26" xml:space="preserve">ſi minùs effectu, ſaltem conatu promovendi, non inboneſta, <lb/>nec ab officio meo aliena videbatur ambitio. </s>
  <s xml:id="echoid-s27" xml:space="preserve">acceſſit tenuis ſpes <lb/>ineſſe bonæ frugis non-nibil, quod &amp; </s>
  <s xml:id="echoid-s28" xml:space="preserve">aliquatenus tibi proſit, <lb/>nec omnino diſpliceat. </s>
  <s xml:id="echoid-s29" xml:space="preserve">Memineris autem obteſtor qui in bis <lb/>literis provectior es, quale ſcriptum attrectas; </s>
  <s xml:id="echoid-s30" xml:space="preserve">non utique <lb/>tibi ſoli elaboratum; </s>
  <s xml:id="echoid-s31" xml:space="preserve">non ſponte productum; </s>
  <s xml:id="echoid-s32" xml:space="preserve">non diuturnà <lb/>meditatione ſubactos exbibens feriantis ingenii conceptus; </s>
  <s xml:id="echoid-s33" xml:space="preserve">at <lb/>Lectiones Scholaſticas; </s>
  <s xml:id="echoid-s34" xml:space="preserve">primùm officii neceſſitate expreſſas; </s>
  <s xml:id="echoid-s35" xml:space="preserve"><lb/>tum ſubinde properantiùs effuſas, ut abſolveretur penſum, ac <lb/>bora deflueret; </s>
  <s xml:id="echoid-s36" xml:space="preserve">demùm ad promiſcui literarii populi inſtructi-<lb/>onem comparatas, cujus intererat complura (qualia tibi vide-<lb/>buntur) leviora non prætermitti; </s>
  <s xml:id="echoid-s37" xml:space="preserve">ut fruſtrà futurus ſis (id <lb/>quod te monitum oportuit, nè multùm expectando tibi pariter <lb/>obſis, ac mibi) accuratum bic quicquam, affabrè poſitum, aut <lb/>concinnè digestum ſperans. </s>
  <s xml:id="echoid-s38" xml:space="preserve">Enimverò, quò tibi ſatisface-<lb/>rem, expediret ſcio multa detruncare, meliora ſubstituere, <lb/>pleraque tranſponere, omnia adincudem limámque revocare; </s>
  <s xml:id="echoid-s39" xml:space="preserve"><lb/>quæ tamen adniti, nec stomacbi mei, nec otii fuit; </s>
  <s xml:id="echoid-s40" xml:space="preserve">ſed nec <lb/>facultatis exequi. </s>
  <s xml:id="echoid-s41" xml:space="preserve">in puris itaque naturalibus (quod aiunt) <lb/>&amp; </s>
  <s xml:id="echoid-s42" xml:space="preserve">prout nata ſunt emittere malui; </s>
  <s xml:id="echoid-s43" xml:space="preserve">quàm operosè lambendo <lb/>aliam in formam, nec ipſam placituram refingere. </s>
  <s xml:id="echoid-s44" xml:space="preserve">quinimò <lb/>poſtquam edendi propoſitum inii, ſeu faſtidio correptus ſeu no-<lb/>vandi ſubiturum ſtudium fugitans, nè quidem borum mag-<lb/>nam partem relegere ſuſtinui; </s>
  <s xml:id="echoid-s45" xml:space="preserve">verùm, quod tenellæ matres
<pb file="0014" n="14" rhead=""/>
factitant, â me depulſum partum amicorum baud recuſan-<lb/>tium nutriciæ curæ commiſi, prout ipſis viſum eſſet, educan-<lb/>dum aut exponendum. </s>
  <s xml:id="echoid-s46" xml:space="preserve">quorum unus (ipſos enim boneſtum <lb/>duco nominatim agnoſcere) D. </s>
  <s xml:id="echoid-s47" xml:space="preserve">Iſaacus Newtonus, collega <lb/>noſter (peregregiæ vir indolis ac inſignis peritiæ) exemplar <lb/>reviſit, aliqua corrigenda monens, ſed &amp; </s>
  <s xml:id="echoid-s48" xml:space="preserve">de ſuo nonnulla <lb/>penu ſuggerens, quæ naſtris alicubi cum laude innexa cernes. <lb/></s>
  <s xml:id="echoid-s49" xml:space="preserve">alter (quem noſtræ gentis baud immeritò Merſennum dix-<lb/>ero, cùm ſuâ tum aliorum operâ provebendis biſce literis na-<lb/>tum) D. </s>
  <s xml:id="echoid-s50" xml:space="preserve">Joh. </s>
  <s xml:id="echoid-s51" xml:space="preserve">Collinſius, ingente ſuo cum labore editionem <lb/>procuravit. </s>
  <s xml:id="echoid-s52" xml:space="preserve">Pθſſem jam alios expectationi tuæ obices ponere, <lb/>ſeu veniæ conciliatrices cauſ@s obtendere (meam ingenii te-<lb/>nuitatem, experimentorum inopiam, alias intercurrentes <lb/>curas) niſi Catonis ſenioris mordaculum illud in me ſubve-<lb/>rerer recaſurum: </s>
  <s xml:id="echoid-s53" xml:space="preserve">_Rectè ſi Amphictyonum decreto con-_ <lb/>_ſtrictus hæc evulgas._ </s>
  <s xml:id="echoid-s54" xml:space="preserve">Hujuſmodi ſaltem præloquium partim <lb/>æquitas exegit, partim in fætum proprium ςοργ@ quædam eli-<lb/>cuit, ut excuſatior is, ac à cenſura munitior prodiret. </s>
  <s xml:id="echoid-s55" xml:space="preserve">ſin <lb/>acrior ſis, nec bæc aure dextrâ admittere velis, pro tuo (per <lb/>me licet) ingenio facias, quantumvìs strenuè reprebendas.</s>
  <s xml:id="echoid-s56" xml:space="preserve"/>
</p>
<pb file="0015" n="15"/>
</div>
<div xml:id="echoid-div8" type="section" level="1" n="8">
<head xml:id="echoid-head11" style="it" xml:space="preserve">Epiſtola; in qua Operis hujus Argumen-<lb/>tum, &amp; ſcopus brevitèr <lb/>exponuntur.</head>
<p>
  <s xml:id="echoid-s57" xml:space="preserve">PErcontaris (amice cum primis chariſſimè) quid in <lb/>Lectionibus iſtis jam prælo ſubditis præſtiterim, aut <lb/>præſtare voluerim. </s>
  <s xml:id="echoid-s58" xml:space="preserve">reſponſo facilè defungi poſſem, <lb/>ea dicendo præſtita videri, quæ ſingularum initia <lb/>pollicentur, è quibus inſequentium methodus, materiâ, <lb/>ſcopus conſtare poterunt ipsâ delibanti. </s>
  <s xml:id="echoid-s59" xml:space="preserve">verùm in ſummam, <lb/>opinor, iſta contrahivis, &amp; </s>
  <s xml:id="echoid-s60" xml:space="preserve">ſub unum aſpectum redigi. </s>
  <s xml:id="echoid-s61" xml:space="preserve">id <lb/>quidem ægrè poſſum, niſi (quod juxtà faſtidioſum ac <lb/>longum eſſet) complura _Tbeoremata_ recitando; </s>
  <s xml:id="echoid-s62" xml:space="preserve">ſed ut-<lb/>cunque morem tibi geram, rerum capita ſuccinctè per-<lb/>ſtringens. </s>
  <s xml:id="echoid-s63" xml:space="preserve">Generatim eò connitor, ut illam, quam tra-<lb/>ctandam ſuſcipio, _Opticæ_ partem aliquatenùs promoveam, <lb/>ejus imprimìs principia explicando; </s>
  <s xml:id="echoid-s64" xml:space="preserve">tum ab ipſis _Vtilia_ <lb/>_Conſectaria_ deducendo; </s>
  <s xml:id="echoid-s65" xml:space="preserve">demùm præcipuos (quos animad-<lb/>verteram) defectus ſupplendo, nec non _vulgatos errores_ <lb/>corrigendo. </s>
  <s xml:id="echoid-s66" xml:space="preserve">huc collimans, ſpeciatim primò receptas hy-<lb/>potheſes ad examen revoco, quatenus admittendæ ſunt <lb/>&amp; </s>
  <s xml:id="echoid-s67" xml:space="preserve">quomodò rectiùs intelligendæ edocere ſtudens; </s>
  <s xml:id="echoid-s68" xml:space="preserve">tum è <lb/>phyſicis ueriſimilibus cauſis ipſas eliciens ac aſtruens. </s>
  <s xml:id="echoid-s69" xml:space="preserve">quâ <lb/>in parte mihi fidei multum attribui nolim; </s>
  <s xml:id="echoid-s70" xml:space="preserve">quæ probabi-<lb/>liora mihi viſa protuli, neutiquam verò talia, quibus ipſe <lb/>magnopere conſidam. </s>
  <s xml:id="echoid-s71" xml:space="preserve">valeant quantum valere poſſunt. <lb/></s>
  <s xml:id="echoid-s72" xml:space="preserve">Saltem hypotheſes ipſas admitti peto, ceu experientiæ <lb/>conſentaneas, nec à ratione quaquàm abhorrentes. </s>
  <s xml:id="echoid-s73" xml:space="preserve"><lb/>Hypotheſibus conſtitutis, ab iis proximè generalia
<pb file="0016" n="16" rhead=""/>
quædam _Tbeoremata_ derivo, partim ab aliis agnita (quæ <lb/>methodi gratiâ, &amp; </s>
  <s xml:id="echoid-s74" xml:space="preserve">propter aliorum probationem, meis de-<lb/>monſtrationibus firmata appono) partim à me obſervata. <lb/></s>
  <s xml:id="echoid-s75" xml:space="preserve">deinad ſpecialia progredior, id mihi negotii ſumens, ut <lb/>_Catoptricæ_, ac _Dioptricæ_ utriuſque, in uſu maximè poſitæ <lb/>(_planæ ſcilicet &amp; </s>
  <s xml:id="echoid-s76" xml:space="preserve">spbæricæ_) potiſſima pertractem _In_ <lb/>_Catoptrica spbærica_ (ſiquidem plana jam olim verè ſatìs, <lb/>ac fusè exculta habetur) ejuſmodi _Tbeoremata_ propono, <lb/>de quibus reflexorum radiorum interſectiones atque limi-<lb/>tes innoteſcunt; </s>
  <s xml:id="echoid-s77" xml:space="preserve">unáque punctorum tam à longè, quàm è <lb/>propinquo radiantium imagines, &amp; </s>
  <s xml:id="echoid-s78" xml:space="preserve">apparentes loci deter-<lb/>minantur; </s>
  <s xml:id="echoid-s79" xml:space="preserve">reſpectu oculi nedum in radiationis axe, ſed <lb/>extra ipſum ubicunque conſtituti. </s>
  <s xml:id="echoid-s80" xml:space="preserve">quæ certè vel nuſquam <lb/>(quod ſciam) aut magnâ ex parte perperàm alibi tractata <lb/>proſtant; </s>
  <s xml:id="echoid-s81" xml:space="preserve">id quod, incidentèr aliorum refutans ſententias, <lb/>cùm ratiociniis perſpicuis, tum experimentis decretoriis <lb/>evictum eo. </s>
  <s xml:id="echoid-s82" xml:space="preserve">_Dioptricam_ porrò tam _planam quàm Spbæ_-<lb/>_ricam_, refractionis noviſſimâ præſtratâ lege vel hypotheſi <lb/>(quam _illustris Carteſius_ detexit, at plerique, reor, melio-<lb/>res _Optici_ jam amplexantur; </s>
  <s xml:id="echoid-s83" xml:space="preserve">quam &amp; </s>
  <s xml:id="echoid-s84" xml:space="preserve">propter aſſignatas <lb/>alicubi rationes veritati conſonam judico) velut à funda-<lb/>mentis extruo. </s>
  <s xml:id="echoid-s85" xml:space="preserve">nec enim eorum, qui principium illud ad-<lb/>miſerunt, ipſum hactenus quiſquam (in ſcriptis intelligo <lb/>quæ viderim luci commendatis) huc applicuit. </s>
  <s xml:id="echoid-s86" xml:space="preserve">Hîc au-<lb/>tem imprimìs puncta radiantia longè diſſita (ſeu quaſi <lb/>parallelos emittentia radios) conſiderans, quo pacto ab <lb/>ipſis proſluentes radii detorquentur exquiro, _Tbeoremata_ <lb/>quædam eliciens, è quibus præcipua _refractorum ſympto_-<lb/>_mata_ liquent, ipſorum interſectiones ac limites dignoſcun-<lb/>tur; </s>
  <s xml:id="echoid-s87" xml:space="preserve">apparentia denique punctorum objectorum loca de-<lb/>ſignantur, tam oculi reſpectu qui in axe, quàm ejus qui <lb/>uſpiam extra axem collocatur. </s>
  <s xml:id="echoid-s88" xml:space="preserve">tunc eadem attento quoad <lb/>puncta ſenſibiliter vicina, ſeu divergentibus radiis allu-<lb/>centia. </s>
  <s xml:id="echoid-s89" xml:space="preserve">ſub extremum, quò paratior ſit horum uſus,
<pb file="0017" n="17" rhead=""/>
punctorum per omnigenas lentes translucentium imagines <lb/>ſingillatim exhibeo determinatas. </s>
  <s xml:id="echoid-s90" xml:space="preserve">Hiſce qualitercunque <lb/>confectis, de magnitudinum dijudicandis (iſtis nempe, <lb/>quæ hujuſmodi conſequuntur inflectiones) apparentiis <lb/>nonnulla generatim attingo; </s>
  <s xml:id="echoid-s91" xml:space="preserve">tum poſteà ſpecialiùs ac <lb/>uberiùs planorum objectorum imagines quales ſunt, &amp; </s>
  <s xml:id="echoid-s92" xml:space="preserve"><lb/>quomodò deſignandæ commonſtro. </s>
  <s xml:id="echoid-s93" xml:space="preserve">ab indè receptui <lb/>cano. </s>
  <s xml:id="echoid-s94" xml:space="preserve">Memoratis autem hiſce paſſim alia πάρερ@α inter-<lb/>ſpergo; </s>
  <s xml:id="echoid-s95" xml:space="preserve">de quibus tu videris, nam eg@ malim reticere.</s>
  <s xml:id="echoid-s96" xml:space="preserve"/>
</p>
<pb file="0018" n="18"/>
  <figure>
    <image file="0018-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0018-01"/>
  </figure>
<p>
  <s xml:id="echoid-s97" xml:space="preserve">Brevitatis gratiâ notæ quædam adhibentur, quarum hîc <lb/>ſubjungitur interpretatio.</s>
  <s xml:id="echoid-s98" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s99" xml:space="preserve">A + B. </s>
  <s xml:id="echoid-s100" xml:space="preserve">_hoc eſt_ A &amp; </s>
  <s xml:id="echoid-s101" xml:space="preserve">B _ſimul acceptæ_.</s>
  <s xml:id="echoid-s102" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s103" xml:space="preserve">A - B. </s>
  <s xml:id="echoid-s104" xml:space="preserve">A, _demptâ_ B.</s>
  <s xml:id="echoid-s105" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s106" xml:space="preserve">A - : </s>
  <s xml:id="echoid-s107" xml:space="preserve">B. </s>
  <s xml:id="echoid-s108" xml:space="preserve">_differentia ipſarun@_ A, &amp; </s>
  <s xml:id="echoid-s109" xml:space="preserve">B.</s>
  <s xml:id="echoid-s110" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s111" xml:space="preserve">A x B. </s>
  <s xml:id="echoid-s112" xml:space="preserve">A _multiplicata, vel ducta in_ B.</s>
  <s xml:id="echoid-s113" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s114" xml:space="preserve">{A/B} - A _diviſa per_ B, _vel applicata ad_ B.</s>
  <s xml:id="echoid-s115" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s116" xml:space="preserve">A = B. </s>
  <s xml:id="echoid-s117" xml:space="preserve">A _æquatur ipſi_ B.</s>
  <s xml:id="echoid-s118" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s119" xml:space="preserve">A &amp;</s>
  <s xml:id="echoid-s120" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s121" xml:space="preserve">B. </s>
  <s xml:id="echoid-s122" xml:space="preserve">A _major eſt quàm_ B.</s>
  <s xml:id="echoid-s123" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s124" xml:space="preserve">A &amp;</s>
  <s xml:id="echoid-s125" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s126" xml:space="preserve">B A _minor eſt quàm_ B.</s>
  <s xml:id="echoid-s127" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s128" xml:space="preserve">A. </s>
  <s xml:id="echoid-s129" xml:space="preserve">B:</s>
  <s xml:id="echoid-s130" xml:space="preserve">: C. </s>
  <s xml:id="echoid-s131" xml:space="preserve">D A _ad_ B _eandem rationem habet, quam_ C _ad_ D.</s>
  <s xml:id="echoid-s132" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s133" xml:space="preserve">A, B, C, D {.</s>
  <s xml:id="echoid-s134" xml:space="preserve">./.</s>
  <s xml:id="echoid-s135" xml:space="preserve">.}. </s>
  <s xml:id="echoid-s136" xml:space="preserve">A, B, C, D _ſunt continuè proportionales_.</s>
  <s xml:id="echoid-s137" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s138" xml:space="preserve">A. </s>
  <s xml:id="echoid-s139" xml:space="preserve">B &amp;</s>
  <s xml:id="echoid-s140" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s141" xml:space="preserve">C. </s>
  <s xml:id="echoid-s142" xml:space="preserve">D. </s>
  <s xml:id="echoid-s143" xml:space="preserve">A _ad_ B _majorem rationem habet, quàm_ C _ad_ D.</s>
  <s xml:id="echoid-s144" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s145" xml:space="preserve">A. </s>
  <s xml:id="echoid-s146" xml:space="preserve">B &amp;</s>
  <s xml:id="echoid-s147" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s148" xml:space="preserve">C. </s>
  <s xml:id="echoid-s149" xml:space="preserve">D. </s>
  <s xml:id="echoid-s150" xml:space="preserve">A _ad_ B _minorcm rationem habet, quàm_ C _ad_ D.</s>
  <s xml:id="echoid-s151" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s152" xml:space="preserve">A. </s>
  <s xml:id="echoid-s153" xml:space="preserve">B + C. </s>
  <s xml:id="echoid-s154" xml:space="preserve">D = \\ &amp;</s>
  <s xml:id="echoid-s155" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s156" xml:space="preserve">\\ &amp;</s>
  <s xml:id="echoid-s157" xml:space="preserve">lt;</s>
  <s xml:id="echoid-s158" xml:space="preserve">} M.</s>
  <s xml:id="echoid-s159" xml:space="preserve">N. </s>
  <s xml:id="echoid-s160" xml:space="preserve">_Rationes_ A ad B, \\ &amp; </s>
  <s xml:id="echoid-s161" xml:space="preserve">C ad D _compoſitæ_ {_adæquant_ \\ _excedunt_ \\ _deficiunt à_} _ratione_ M \ȧd N.</s>
  <s xml:id="echoid-s162" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s163" xml:space="preserve">Aq. </s>
  <s xml:id="echoid-s164" xml:space="preserve">_Quadratum ex_ A.</s>
  <s xml:id="echoid-s165" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s166" xml:space="preserve">√ A. </s>
  <s xml:id="echoid-s167" xml:space="preserve">_Latus, vel radix quadrata ipſius_ A.</s>
  <s xml:id="echoid-s168" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s169" xml:space="preserve">A c. </s>
  <s xml:id="echoid-s170" xml:space="preserve">_Cubus e x _ A.</s>
  <s xml:id="echoid-s171" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s172" xml:space="preserve">√ Aq + Bq. </s>
  <s xml:id="echoid-s173" xml:space="preserve">_Latus compoſiti ex_ Aq &amp; </s>
  <s xml:id="echoid-s174" xml:space="preserve">Bq.</s>
  <s xml:id="echoid-s175" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s176" xml:space="preserve">Reliquas, ſi quæ occurrunt, abbreviaturas Lector facili conjectur &amp;</s>
  <s xml:id="echoid-s177" xml:space="preserve"><unsure/> <lb/>capiet, præſerti@ in analyſi tantillùm verſatus.</s>
  <s xml:id="echoid-s178" xml:space="preserve"/>
</p>
<pb o="1" file="0019" n="19"/>
</div>
<div xml:id="echoid-div9" type="section" level="1" n="9">
<head xml:id="echoid-head12" xml:space="preserve">Lect. I.</head>
<p>
  <s xml:id="echoid-s179" xml:space="preserve">I. </s>
  <s xml:id="echoid-s180" xml:space="preserve">PRæfatorio jam vinculo ſolutus, &amp; </s>
  <s xml:id="echoid-s181" xml:space="preserve">ſcopulum præterve-<lb/>
<anchor type="note" xlink:label="note-0019-01a" xlink:href="note-0019-01"/>
ctus Rhetoricum, ad muneris mei proprium opus ac-<lb/>cingor. </s>
  <s xml:id="echoid-s182" xml:space="preserve">Imprimis autem novi quod inîerim conſilii <lb/>rationem, paucis expediam. </s>
  <s xml:id="echoid-s183" xml:space="preserve">Cum prius inſtitutum ur-<lb/>gens adverterim, occurrere pleraque nimiam attentio-<lb/>nem deſiderantia, nec ex improviſo auſcultantibus, in-<lb/>dè ſatis opportuna; </s>
  <s xml:id="echoid-s184" xml:space="preserve">incommodum etiam illud à puram Geometriam <lb/>attrectantibus haud poſſe declinari; </s>
  <s xml:id="echoid-s185" xml:space="preserve">conſtitui, derelictâ tantiſper <lb/>iſtâ, protinus in amæniores (floribus nempe Phyſicis depicto, &amp; </s>
  <s xml:id="echoid-s186" xml:space="preserve"><lb/>fructibus conſitos Mechanicis) mixtæ quam appellitant Matheſeos <lb/>Campos deviare; </s>
  <s xml:id="echoid-s187" xml:space="preserve">Opticæ nimirum, Mechanicæ, Coſmographiæ, re-<lb/>liquæ cujuſcunque, prout occaſio feret, &amp; </s>
  <s xml:id="echoid-s188" xml:space="preserve">commodum videbitur. </s>
  <s xml:id="echoid-s189" xml:space="preserve">Ne-<lb/>que tamen animus erit ullius ex his longè diffuſa latifundia pervagari, <lb/>vel extremos fines circumire; </s>
  <s xml:id="echoid-s190" xml:space="preserve">ſed ad ejus quaſi metropolim è veſtigio <lb/>rectà procedere; </s>
  <s xml:id="echoid-s191" xml:space="preserve">primas tantum hypotheſes excutere, præcipuáque <lb/>(quibus illa tam vaſta theorematum moles incumbit) fundamenta de-<lb/>nudare; </s>
  <s xml:id="echoid-s192" xml:space="preserve">tum verò nonnulla, palmaria quidem illa, ſtatim emergentia <lb/>corollaria ſubtexere. </s>
  <s xml:id="echoid-s193" xml:space="preserve">Quorum certè σκέψις jucunda praſ@r@im, utilis, <lb/>&amp; </s>
  <s xml:id="echoid-s194" xml:space="preserve">fructuoſa videri poteſt; </s>
  <s xml:id="echoid-s195" xml:space="preserve">quum è principiis rectè poſitis, probéque <lb/>perceptis reliquorum &amp; </s>
  <s xml:id="echoid-s196" xml:space="preserve">firma fides, &amp; </s>
  <s xml:id="echoid-s197" xml:space="preserve">facilis comprehenſio ſub-<lb/>naſcantur.</s>
  <s xml:id="echoid-s198" xml:space="preserve"/>
</p>
<div xml:id="echoid-div9" type="float" level="2" n="1">
<note position="right" xlink:label="note-0019-01" xlink:href="note-0019-01a" xml:space="preserve">_Præceſſerat an_-<lb/>_teloquium oc_-<lb/>_caſioni, quæ_ <lb/>_fui@, adap@a_-<lb/>_tum_.</note>
</div>
<p>
  <s xml:id="echoid-s199" xml:space="preserve">II. </s>
  <s xml:id="echoid-s200" xml:space="preserve">Ab Optica ſumemus exordium; </s>
  <s xml:id="echoid-s201" xml:space="preserve">ſcientia cum primis Nobili; <lb/></s>
  <s xml:id="echoid-s202" xml:space="preserve">quam cum peculiaris amænitas, tum ingens commendat utilitas. </s>
  <s xml:id="echoid-s203" xml:space="preserve">Nam <lb/>Naturæ ſimul detegendis arcanis, ac explicandis Phænomenis minimè <lb/>vos latet quantopere conducat; </s>
  <s xml:id="echoid-s204" xml:space="preserve">neque minus ad Aſtronomicas rationes <lb/>quàm planè neceſſaria ſit; </s>
  <s xml:id="echoid-s205" xml:space="preserve">ut Perſpectivam, Picturam, &amp; </s>
  <s xml:id="echoid-s206" xml:space="preserve">his agnatas <lb/>alias eximias Artes taceam; </s>
  <s xml:id="echoid-s207" xml:space="preserve">quæ quantæ quantæ ſunt ab ea pendent, <lb/>ac prineipia ſua mutuantur. </s>
  <s xml:id="echoid-s208" xml:space="preserve">Ut &amp; </s>
  <s xml:id="echoid-s209" xml:space="preserve">præteream qualia, certè vix pretio
<pb o="2" file="0020" n="20" rhead=""/>
ſuo æſtimanda, ad vitæ communis uſum beneſicia ſubminiſtret; </s>
  <s xml:id="echoid-s210" xml:space="preserve">visûs <lb/>imperfectionibus &amp; </s>
  <s xml:id="echoid-s211" xml:space="preserve">vitiis tam prompta, quàm certa, minimi ſumptûs, <lb/>&amp; </s>
  <s xml:id="echoid-s212" xml:space="preserve">nullius periculi remedia conferendo. </s>
  <s xml:id="echoid-s213" xml:space="preserve">Neque, quum curioſiſſimus <lb/>iſte ſenſus noſter ità varias indies, ità miras rerum ſpecies exhibeat <lb/>nobis; </s>
  <s xml:id="echoid-s214" xml:space="preserve">non admodum oblectare nos, non eximiâ voluptate mentes <lb/>noſtras afficere poſſit, unde talis emergat apparentiarum diverſitas, &amp; </s>
  <s xml:id="echoid-s215" xml:space="preserve"><lb/>quis ſit illas attingendi modus nedum accuratè, certóque cognoſcere, <lb/>ſed utcunque veriſimilitér arbitrari; </s>
  <s xml:id="echoid-s216" xml:space="preserve">præſertim quum in nullâ parte <lb/>noſtri, nec in tota fortaſſis rerum compage, neceſſitatibus, commo-<lb/>dis, &amp; </s>
  <s xml:id="echoid-s217" xml:space="preserve">voluptatibus noſtris proſpicientis melioris naturæ ſeu fines <lb/>agendi, ſeu modos pleniùs queamus perſpicere; </s>
  <s xml:id="echoid-s218" xml:space="preserve">nuſquam adeò di-<lb/>ftinctiùs aut apertiùs opificis _πανσφδ_ eluceat artificium. </s>
  <s xml:id="echoid-s219" xml:space="preserve">Verùm <lb/>elogia pertexere non vacat, aut convenit nobis. </s>
  <s xml:id="echoid-s220" xml:space="preserve">Rem potiùs ipſam <lb/>aggrediamur.</s>
  <s xml:id="echoid-s221" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s222" xml:space="preserve">III. </s>
  <s xml:id="echoid-s223" xml:space="preserve">Quæ circa viſum occupatur diſcipiina communiter in tria mem-<lb/>bra diſpertitur; </s>
  <s xml:id="echoid-s224" xml:space="preserve">primum, quod viſus directis radiis objecta cernentis <lb/>affectiones conſiderat (hoc ſpeciatim Optice nominatur;) </s>
  <s xml:id="echoid-s225" xml:space="preserve">alterum, <lb/>quod è radiorum ab opacis corporibus repercuſſu oriundas ſpeculatur <lb/>apparentias (cui Catoptricæ nomen inditum;) </s>
  <s xml:id="echoid-s226" xml:space="preserve">tertium denique, quod <lb/>ideò Dioptrica vocitatur, quia cauſas inveſtigat, aut exponit eorum <lb/>quæ à radiis apparent per diverſa media tranſlucentibus, &amp; </s>
  <s xml:id="echoid-s227" xml:space="preserve">eorum oc-<lb/>curſu demutatis Quam diſtributionem ut non improbamus, ità nobis <lb/>haud obſervandam proponimus; </s>
  <s xml:id="echoid-s228" xml:space="preserve">nedum quia multa pariter his com-<lb/>munia ſunt, at præcipuè quia viſio quævis, ut libet ſimplex ac directa, <lb/>ſicuti reverà non abſque nonnulla radiorum inflectione peragitur, ità <lb/>nec eâ ſeclusâ penitus intelligi poteſt aut explicari. </s>
  <s xml:id="echoid-s229" xml:space="preserve">Igitur hujuſmodi <lb/>methodo potiùs inſiſtendum cenſemus; </s>
  <s xml:id="echoid-s230" xml:space="preserve">ut nempe primò viſionis cau-<lb/>ſas (quæ ſcilicet illam extrinſecus eſſiciunt, aut afficiunt) examine-<lb/>mus; </s>
  <s xml:id="echoid-s231" xml:space="preserve">tum ut videndi modum (hoc eſt quo pacto ſenſus hic noſter ido-<lb/>neis organis inſtructus iſtis concurrentibus cauſis, objectorum illas, <lb/>quas experimur, differentias apprehendit) adnitamur exponere; </s>
  <s xml:id="echoid-s232" xml:space="preserve">de-<lb/>hinc, ut Phænomena quædam ſelectiora ſuſcipiamus elucidanda; <lb/></s>
  <s xml:id="echoid-s233" xml:space="preserve">poſtremóque forſan, ut de viſùs remediis ac ſubſidiis aliquid ſub-<lb/>jungamus.</s>
  <s xml:id="echoid-s234" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s235" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s236" xml:space="preserve">Viſionis cauſas externas quod attinet, nemini jam dubium eſt, <lb/>exiſtimo, non ullâ (quanquam _Empedocli, Platoui, Enclidi_, veteribus <lb/>aliisid placitum erat) ab oculo radiorum emiſſione, verùm ab objectis <lb/>defluente re quâpiam, oculoſque percellente viſum effici; </s>
  <s xml:id="echoid-s237" xml:space="preserve">quod &amp;</s>
  <s xml:id="echoid-s238" xml:space="preserve">
<pb o="3" file="0021" n="21" rhead=""/>
_Democrito_ jam olim, ejúſque ſequaci (dicam an ſimio? </s>
  <s xml:id="echoid-s239" xml:space="preserve">) ſuboluer@t <lb/>_Epicu<unsure/>ro_. </s>
  <s xml:id="echoid-s240" xml:space="preserve">Quod ſanè malim adſumere, vel ſupponere, quam poſt tot <lb/>alios operoſo niſu comprobare. </s>
  <s xml:id="echoid-s241" xml:space="preserve">Certè (quo breviſſimè tangam hanc <lb/>quæſtionem) ſic in alia qualibet evenit ſenſione, (quidni pariter in vi-<lb/>ſu?) </s>
  <s xml:id="echoid-s242" xml:space="preserve">non ut ſenſus in objecta feratur, ſed ut ipſa ſe ſenſibus imprimant; <lb/></s>
  <s xml:id="echoid-s243" xml:space="preserve">immediato nempe contactu, vel medii cujuſdam ſeu projecti, ſeu com-<lb/>moti interventu. </s>
  <s xml:id="echoid-s244" xml:space="preserve">Tum ratio vetat, ut ex ocello quicquam in immenſam <lb/>adeò circumquaque diſtantiam credamus emanare; </s>
  <s xml:id="echoid-s245" xml:space="preserve">neque quod ſic <lb/>emanet in eo quidpiam aptum natum deprehendimus. </s>
  <s xml:id="echoid-s246" xml:space="preserve">Totus enim-<lb/>vero pellucidis humoribus aut membranulis conſtat opacis, ad tranſ-<lb/>mittendam lucem, vel ad eam excipiendam, aptiſſimè, ſed ad progig-<lb/>nendam à ſe vel ejaculandam haudquaquam comparatis. </s>
  <s xml:id="echoid-s247" xml:space="preserve">Quòd ſi lu-<lb/>cem ipſe profunderet, inſitiſque radiis attingeret objecta, quidniden-<lb/>ſiſſimis in tenebris hoc præſertim faceret, &amp; </s>
  <s xml:id="echoid-s248" xml:space="preserve">feles vel (Hiſtoricis ſi <lb/>placet) _Tiberii_ fieremus omnes? </s>
  <s xml:id="echoid-s249" xml:space="preserve">Quæ, dico, lucis externæ tam indiſ-<lb/>penſabilis ad viſum neceſſitas eſſet? </s>
  <s xml:id="echoid-s250" xml:space="preserve">συνωγας equidem Platonicæ. </s>
  <s xml:id="echoid-s251" xml:space="preserve"><lb/>Sonum audio, vim non capio. </s>
  <s xml:id="echoid-s252" xml:space="preserve">Demum ab objectis, etiam à tergo ſitis, <lb/>circumfuſas ſpecies quas vocant, ad oculos deportari, ſuíque percep-<lb/>tionem efficere, cùm à ſpeculis, tum ab aliis innumeris perquam ob-<lb/>viis experimentis compertum habetur; </s>
  <s xml:id="echoid-s253" xml:space="preserve">illarum igitur eſſicaciæ quidni <lb/>commodiſſimè viſionem adſcribamus? </s>
  <s xml:id="echoid-s254" xml:space="preserve">ſatìs hæc illam quam adſumimus <lb/>vulgarem jam hypotheſin adſtruunt, quam &amp; </s>
  <s xml:id="echoid-s255" xml:space="preserve">totus dicendorum tenor <lb/>luculentè confirmabit.</s>
  <s xml:id="echoid-s256" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s257" xml:space="preserve">V. </s>
  <s xml:id="echoid-s258" xml:space="preserve">Cùm verò multa viſum aſſiciant diverſimode, puta lux, lumen, <lb/>dies, crepuſculum, colores,<unsure/> rerum imagines, phaſmata; </s>
  <s xml:id="echoid-s259" xml:space="preserve">nec tamen <lb/>abſque luce. </s>
  <s xml:id="echoid-s260" xml:space="preserve">(Præſente nimirum aut prævia) quidvis horum aliquid <lb/>peragat, perſpicuum eſt lucis hîc præcipuas partes, primariam effica-<lb/>ciam fore. </s>
  <s xml:id="echoid-s261" xml:space="preserve">Quinimo rem ſedulò penſitantes, eò deveniemus, opinor, <lb/>ut varias his omnibus adnexas apparentias non aliunde quàm ex diverſi-<lb/>modâ lucis unius operatione putemus proficiſci. </s>
  <s xml:id="echoid-s262" xml:space="preserve">Cùm nempe lux ſit <lb/>illud quicquid ſit quod à corpore lucido (quale ſtella, ignis, flamma) <lb/>proveniens immediatè viſum afficit, lumen nil videtur aliud quàm lux <lb/>in corpuſcula quædam opaca (ſeu lucem non penitùs excipientia) <lb/>πς ωειΕΧονη interſperſa impingens, nec non ab iis in omnes undique <lb/>partes reſiliens; </s>
  <s xml:id="echoid-s263" xml:space="preserve">quæ ſcilicet in oculum itineri ſuo expoſitum tumultu-<lb/>ariè delapſa confuſam quandam apparentiam excitat; </s>
  <s xml:id="echoid-s264" xml:space="preserve">quàm, ſi fortior <lb/>ſit, eique prorogandæ lucens præſtò ſit, appellamus diem; </s>
  <s xml:id="echoid-s265" xml:space="preserve">at ſi de-<lb/>bilior fuerit, ejúſque fons abſceſſerit, crepuſculum dicimus. </s>
  <s xml:id="echoid-s266" xml:space="preserve">Etiam <lb/>color nil fermè videtur aliud, quam lux à corporibus quibus occurrit
<pb o="4" file="0022" n="22" rhead=""/>
majuſculis, &amp; </s>
  <s xml:id="echoid-s267" xml:space="preserve">aliquatenus ſtabilem ſuarum partium ſitum retinentibus <lb/>(pro varia particularum, è quibus illa componuntur, figura, diſpoſiti-<lb/>one, textura, hoc vel illo modo) detorta, vel utcunque repercuſſa; <lb/></s>
  <s xml:id="echoid-s268" xml:space="preserve">nimirum ut ejuſmodi corporibus illapſa lux vel motu ſuo, vel agendi <lb/>virtute, vel ipſâ quantitate ſuâ (quoad raritatem intelligo, vel denſita-<lb/>tem; </s>
  <s xml:id="echoid-s269" xml:space="preserve">radiorum copiam, aut paucitatem) talis evadat, &amp; </s>
  <s xml:id="echoid-s270" xml:space="preserve">pro modi <lb/>diſcrimine diſpares procreet apparentias, à quibus eam variis colorum <lb/>nominibus inſignimus. </s>
  <s xml:id="echoid-s271" xml:space="preserve">Imagines autem nil planè ſunt aliud, quum <lb/>lux ab objectis ita reflexa, vel refracta, ut rurſus in unum locum, ta-<lb/>lémque recolligatur ſitum, qualem tunc obtinuit, quum ab originali <lb/>proflueret objecto; </s>
  <s xml:id="echoid-s272" xml:space="preserve">directóque verſus oculum itinere procederet; </s>
  <s xml:id="echoid-s273" xml:space="preserve">quo <lb/>fit ut ſimiliter objecta, ſed tanquam alibi collocata repræſentent. </s>
  <s xml:id="echoid-s274" xml:space="preserve"><lb/>Phaſmata deníque ſunt imaginum quaſi colores, pro lucis diverſa me-<lb/>dia trajicientis alia ac alia quoad motum, vim, quantitatem affectione <lb/>diverſa variati Craſſiuſculè jam iſta proponimus; </s>
  <s xml:id="echoid-s275" xml:space="preserve">quorum forſan ali-<lb/>qua ſaltem in dicendorum progreſſu magis eluceſcent.</s>
  <s xml:id="echoid-s276" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s277" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s278" xml:space="preserve">Cùm itáque lux in viſione peragenda, diverſiſque procreandis <lb/>apparentiis ità quaſi paginam utramque faciat; </s>
  <s xml:id="echoid-s279" xml:space="preserve">Et reverà præter illam <lb/>nil aliud ſenſum ingredi, vel commovere videatur; </s>
  <s xml:id="echoid-s280" xml:space="preserve">de illa primo diſpi-<lb/>ciendum venit. </s>
  <s xml:id="echoid-s281" xml:space="preserve">Et ejuſce quidem de natura à Phyſicis magnoperè <lb/>deſceptatur; </s>
  <s xml:id="echoid-s282" xml:space="preserve">an puta ſit corporea quædam ſubſtantia, an qualitas; </s>
  <s xml:id="echoid-s283" xml:space="preserve">an <lb/>actio tantùm, aut motus quidam; </s>
  <s xml:id="echoid-s284" xml:space="preserve">de productione quoque conſequen-<lb/>tèr ejuſdem, &amp; </s>
  <s xml:id="echoid-s285" xml:space="preserve">propagatione diſquiritur, utrùm continuo per medium <lb/>tranſitu, vel medii duntaxat impulſu, vel ſuâ ipſius multiplicatione <lb/>quâdam huc propagetur; </s>
  <s xml:id="echoid-s286" xml:space="preserve">quales ego quæſtiones curioſè non eventilabo. <lb/></s>
  <s xml:id="echoid-s287" xml:space="preserve">Quod iſtam ſaltem ſententiam attinet, quæ lucem accidentium claſſi <lb/>accenſet; </s>
  <s xml:id="echoid-s288" xml:space="preserve">quando veris corporeis effectibus quales ſunt rectâ progredi, <lb/>repercnti, refringi, calorem excitare, ſenſum afficere) veræ ſubliſten-<lb/>tes cauſæ, veri locales motus aſſignari debeant; </s>
  <s xml:id="echoid-s289" xml:space="preserve">neque quomodò meræ <lb/>qualitati, vel accidenti cuipiam iſta competant intelligere mihi datum <lb/>ſit; </s>
  <s xml:id="echoid-s290" xml:space="preserve">quinetiam quo ſeſe pacto multiplicare valeat, id genus entium, <lb/>quâ ratione vim ullam exerere, cùm è cordatioribus &amp; </s>
  <s xml:id="echoid-s291" xml:space="preserve">rerum intima <lb/>perſerutantibus Philoſophis haud pauci ſe parùm capere profiteantur; </s>
  <s xml:id="echoid-s292" xml:space="preserve"><lb/>eam haud dubitem hîc miſſam facere. </s>
  <s xml:id="echoid-s293" xml:space="preserve">Verùm an corporeæ quædam <lb/>άπόῤῥοι{αι}, de lucidi corporis viſceribus emanantes, totúmque nobis &amp; </s>
  <s xml:id="echoid-s294" xml:space="preserve"><lb/>ipſi interjectum ſpatium quàm perniciſſimè tranſcurrentes lucem con-<lb/>ſtituant; </s>
  <s xml:id="echoid-s295" xml:space="preserve">vel an illa potius nihil ſit aliud quàm ipſius lucentis actio, <lb/>contigua ſibi corpora prementis ac impellentis, iíſque mediantibus <lb/>alia, quæ adjacent; </s>
  <s xml:id="echoid-s296" xml:space="preserve">tum &amp; </s>
  <s xml:id="echoid-s297" xml:space="preserve">horum interceſſu rurſus alia proximè ſuc-
<pb o="5" file="0023" n="23" rhead=""/>
cedentia; </s>
  <s xml:id="echoid-s298" xml:space="preserve">nec non ità perpetuâ deinceps ad nos deductâ ſerie; </s>
  <s xml:id="echoid-s299" xml:space="preserve">vix au-<lb/>ſim certè mihi dijudicandum accipere; </s>
  <s xml:id="echoid-s300" xml:space="preserve">adeò paribus u@raque pars ar-<lb/>gumentis niti videtur, æquis utraque difficultatibus urgeri. </s>
  <s xml:id="echoid-s301" xml:space="preserve">Quin eo <lb/>fere propendeo, ut cenſeam utroque ſubinde modo lucem procreari, <lb/>tam per effluvia corporea, quàm per continuum impulſum; </s>
  <s xml:id="echoid-s302" xml:space="preserve">ſatiúſque <lb/>fore nonnullos ejus effectus huic, alios illi tribuere. </s>
  <s xml:id="echoid-s303" xml:space="preserve">Sanè cùm ad <lb/>quantum intervallum undiquaque protenſum exiguæ lampadis flam-<lb/>mula ſe vividè conſpiciendam præbeat, adeò quidem ut integrum ejus <lb/>radiatione circumpolitum medium perfundi compleríque videatur, <lb/>animadverto; </s>
  <s xml:id="echoid-s304" xml:space="preserve">quomodo tantillum corpus tali tamdiu ſuppeditandæ <lb/>profluviorum copiæ par ſit; </s>
  <s xml:id="echoid-s305" xml:space="preserve">quomodo dum ea profundit non ipſum <lb/>pluſquam exhauriatur, &amp; </s>
  <s xml:id="echoid-s306" xml:space="preserve">confeſtim evaneſcat, haud facile capio. <lb/></s>
  <s xml:id="echoid-s307" xml:space="preserve">Cum verò rurſus lucis inflectiones, illáſque qui conſequuntur effectus <lb/>cogito, vix animo meo nudus impulſus facit ſatis. </s>
  <s xml:id="echoid-s308" xml:space="preserve">Itaque mentis anxi-<lb/>us hæreo. </s>
  <s xml:id="echoid-s309" xml:space="preserve">Veruntamen quia de natura lucis aliquid præſternam expe-<lb/>dit, iis quas mox tradam hypotheſibus nonnihil explicandis congruum; </s>
  <s xml:id="echoid-s310" xml:space="preserve"><lb/>hoc ſe modo, vel non abſimili rem habere concipio.</s>
  <s xml:id="echoid-s311" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s312" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s313" xml:space="preserve">Pono corpus omne lucidum, ut tale, congeriem eſſe quandam <lb/>corpuſculorum ultrà pene quàm cogitari poteſt minutorum &amp; </s>
  <s xml:id="echoid-s314" xml:space="preserve">exilium; <lb/></s>
  <s xml:id="echoid-s315" xml:space="preserve">horum autem unumquodque vehementiſſimo motu percitum, aliquò <lb/>(ſecundum legem iſtam naturæ ſatìs receptam &amp; </s>
  <s xml:id="echoid-s316" xml:space="preserve">exploratam) rectà <lb/>tendere; </s>
  <s xml:id="echoid-s317" xml:space="preserve">tum medium circumſtare, fluidum quoque (cujus nempe <lb/>partes nullo colligatæ nexu quaquaverſùm liberè feruntur) è corpori-<lb/>bus aggregatum, exiliſſimis quidem &amp; </s>
  <s xml:id="echoid-s318" xml:space="preserve">illis, aſt priorum reſpectu bene <lb/>craſſis &amp; </s>
  <s xml:id="echoid-s319" xml:space="preserve">ſolidis; </s>
  <s xml:id="echoid-s320" xml:space="preserve">ità tamen ut hoc meatus habeat, &amp; </s>
  <s xml:id="echoid-s321" xml:space="preserve">interſtitia tenui-<lb/>oribus illis admittendis opportuna; </s>
  <s xml:id="echoid-s322" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s323" xml:space="preserve">horum craſſiorum corpuſ-<lb/>culorum occurſu progreſſum impediri multorum ex illis, quæ in lucidi <lb/>ſuperficie verſantur, aut ab ea ruunt corpuſculis; </s>
  <s xml:id="echoid-s324" xml:space="preserve">ut neceſſe ſit iis ſic <lb/>inhibitis, atque repulſis introrſum ſe recipere, quo fit ut dicta conge-<lb/>ries (aliis etiam in eam aliunde confluentibus ejuſdem naturæ corpuſcu-<lb/>lis) aliquatenus intra ſuos cancellos reſtringatur, nec toto ſtatim in au-<lb/>ras expanſa diſſipetur. </s>
  <s xml:id="echoid-s325" xml:space="preserve">Interim verò complura per dictos canales re-<lb/>pertâ viâ curſum ſuum rectà continuare, materiam inibi deprehenſam <lb/>haud ita foniter obſiſtentem in fugam agentia, &amp; </s>
  <s xml:id="echoid-s326" xml:space="preserve">ante ſe protrudentia; </s>
  <s xml:id="echoid-s327" xml:space="preserve"><lb/>quorum veſtigiis alia de lucido corpore ſimiliter prodeuntia prorſus <lb/>inſiſtent, longúmque ſimul omnia lucis rivulum efficient, indeflexâ <lb/>ſerie procurrentem. </s>
  <s xml:id="echoid-s328" xml:space="preserve">Quin &amp; </s>
  <s xml:id="echoid-s329" xml:space="preserve">iſtorum fortè nonnulla memoratas me-<lb/>dii craſſiores particulas impetu ferire tam prævalido, nonnunquam ut <lb/>ipſas quoque cedere cogant, &amp; </s>
  <s xml:id="echoid-s330" xml:space="preserve">ſecum conſpirantes in directum adja-
<pb o="6" file="0024" n="24" rhead=""/>
centia, corpora propellere; </s>
  <s xml:id="echoid-s331" xml:space="preserve">quæ &amp; </s>
  <s xml:id="echoid-s332" xml:space="preserve">pari modo proximè ſuccedentibus <lb/>vim inferent, &amp; </s>
  <s xml:id="echoid-s333" xml:space="preserve">ita continuò, ſic ut ſimul &amp; </s>
  <s xml:id="echoid-s334" xml:space="preserve">ſemel indefinitè protenſa <lb/>talium corpuſculorum ſeries promoveatur, &amp; </s>
  <s xml:id="echoid-s335" xml:space="preserve">antrorſum connitatur; <lb/></s>
  <s xml:id="echoid-s336" xml:space="preserve">qualis utrolibet modo producta lucis propago radius conſuevit appella-<lb/>ri. </s>
  <s xml:id="echoid-s337" xml:space="preserve">Ità quidem rem exiſtimo ſimpliciter obtingere, donec medìum <lb/>permanet homogeneum, hoc eſt ejuſdem fermè magnitudinis, ſolidi-<lb/>tatis, ac figuræ partibus conſtans, &amp; </s>
  <s xml:id="echoid-s338" xml:space="preserve">ſimilibus interſtitiis pervium; </s>
  <s xml:id="echoid-s339" xml:space="preserve">at <lb/>ſi medium occurrat aliter affectum, è diverſis quippe ſecundum quan-<lb/>titatem aut figuram particulis compactum, poriſque laxioribus, aut <lb/>ſtrictioribus pertuſum, cujúſque proinde materia vel promptius cedat, <lb/>aut contumaciùs obluctetur, oportebit illius ſeu curſus, ſeu impulſùs <lb/>vim, effectúmque demutari; </s>
  <s xml:id="echoid-s340" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s341" xml:space="preserve">ſi novi medii ſuperficies ita tran-<lb/>ſeunti lucis amni ſe obliquam objiciat, ejus quoque directionem infrin-<lb/>gi; </s>
  <s xml:id="echoid-s342" xml:space="preserve">vel ἀνακλασιν contingere, quam _Ariftoteles_ vocat, eo nomine <lb/>(quas nunc diſtinguere ſolemus) reflectionem ſimul ac refractionem <lb/>complectens. </s>
  <s xml:id="echoid-s343" xml:space="preserve">Enimverò materiæ impingens ità compactæ, ut venienti <lb/>tranſi um perneget, aut prementis impetum inconcuſſa ſuſtineat aliò tota <lb/>quò facillimè poterit &amp; </s>
  <s xml:id="echoid-s344" xml:space="preserve">directiſſimè, regredietur &amp; </s>
  <s xml:id="echoid-s345" xml:space="preserve">reſiliet; </s>
  <s xml:id="echoid-s346" xml:space="preserve">aliò vim <lb/>ſuam quàm retinet omnem derivabit; </s>
  <s xml:id="echoid-s347" xml:space="preserve">id quod lucis reflectio dicitur <lb/>(Hujuſmodi verò corpus lucem non ſuſcipiens eatenus opacum, (Hoc <lb/>eſt terrenum, ut Grammatici volunt, ab Opevocabulo priſco tellurem <lb/>deſignante) appellatur; </s>
  <s xml:id="echoid-s348" xml:space="preserve">quatenus autem ſibimet incurrentem aliò <lb/>projicit, illuſtratum dici; </s>
  <s xml:id="echoid-s349" xml:space="preserve">quatenus objecti ſpeciem redhibet aſpicienti, <lb/>ſpeculum.) </s>
  <s xml:id="echoid-s350" xml:space="preserve">Quòd ſi verò materia luci progredienti ſic obviam facta <lb/>tranſitum utcunque præbeat, ejuſve conatum excipiat, lentiùs tamen <lb/>aut paratiùs præ illa, per quam priùs decurrebat, tum virtutis ſuæ <lb/>quantitate aliquantùm hinc variatâ ſimul à recto quod affectabat itinere <lb/>deflectetur; </s>
  <s xml:id="echoid-s351" xml:space="preserve">eo nimirum ordine modóque quem poſthac conabimur <lb/>elicere. </s>
  <s xml:id="echoid-s352" xml:space="preserve">Qualis effectus refractionis nomine venire ſolet(ſubnotetur <lb/>autem, hoc modo lucem intromittens medium eatenus perſpicuum, <lb/>diaphanum, tranſparens, pellucidum appellari.) </s>
  <s xml:id="echoid-s353" xml:space="preserve">Ità lucis naturam, <lb/>originem, propagationem, ac progreſſum ὀλχερῶς (omiſſis quæ adjun-<lb/>gi poſſent pleriſque minùs ad noſtrum propoſitum ſpectantibus) expo-<lb/>no; </s>
  <s xml:id="echoid-s354" xml:space="preserve">nec aliud ferè præter hæc requiro declarandis hypotheſibus, quos <lb/>communiter adſumunt Optici; </s>
  <s xml:id="echoid-s355" xml:space="preserve">quæque neceſſariò debent huic extru-<lb/>endæ Scientiæ præſterni. </s>
  <s xml:id="echoid-s356" xml:space="preserve">Comprobandis autem iis, quæ dixi non in-<lb/>cumbam; </s>
  <s xml:id="echoid-s357" xml:space="preserve">cùm &amp; </s>
  <s xml:id="echoid-s358" xml:space="preserve">(quod inſtituto noſtro ſatis eſt) talia dari poſſe non <lb/>minùs ipſâ luce clarum videatur, imò reverà dari complura declarent <lb/>experimenta. </s>
  <s xml:id="echoid-s359" xml:space="preserve">Opticas verò quas innui hypotheſes præcipuas ſubjunge-<lb/>mus, &amp; </s>
  <s xml:id="echoid-s360" xml:space="preserve">nonnihil attentabimus explicare.</s>
  <s xml:id="echoid-s361" xml:space="preserve"/>
</p>
<pb o="7" file="0025" n="25" rhead=""/>
<p>
  <s xml:id="echoid-s362" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s363" xml:space="preserve">1. </s>
  <s xml:id="echoid-s364" xml:space="preserve">Radii lucis (hoc eſt lucidi tranſitûs aut impulſùs quales de-<lb/>ſcripſimus tramites) in eodem exiſtentes ſimilari medio directi ſunt. <lb/></s>
  <s xml:id="echoid-s365" xml:space="preserve">Hoc è dictis abunde patet. </s>
  <s xml:id="echoid-s366" xml:space="preserve">Quin indè Corollarii vice deducitur radios <lb/>quoad rem ipſam, Phyſicéque loquendo figurà priſmaticos eſſe, vel <lb/>cylindricos. </s>
  <s xml:id="echoid-s367" xml:space="preserve">Nempe corpuſculum illud quodpiam in lucidi ſuperficie <lb/>poſitum, à quo radius originem ſuam ducit, dum à primò ſuo loco ceu <lb/>baſe defertur aut totâ ſuâ ſuperficie contiguum ſibi corpus rectà propel-<lb/>lit, figuræ ſuæ (vel impulſi ſaltem corporis figuræ) congruum deſig-<lb/>nat, ſuper hac vel illa baſe conſtitutum, ſolidum longum, exile, teres, <lb/>quale cylindrus, aut priſma. </s>
  <s xml:id="echoid-s368" xml:space="preserve">Proinde quando Mathematicè rem tra-<lb/>ctamus, iſtos radios pro rectis lineis habere poſſumus; </s>
  <s xml:id="echoid-s369" xml:space="preserve">tum quia reve-<lb/>ra ſunt adeo tenues &amp; </s>
  <s xml:id="echoid-s370" xml:space="preserve">recti; </s>
  <s xml:id="echoid-s371" xml:space="preserve">tum quia plerumque pro cylindricis ejuſ-<lb/>modi ſeu priſmaticis figuris ipſarum axes ità ſumi poſſunt, ut nihil indè <lb/>ratiocinio Mathematico derogetur.</s>
  <s xml:id="echoid-s372" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s373" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s374" xml:space="preserve">2. </s>
  <s xml:id="echoid-s375" xml:space="preserve">Ab omni corporis lucidi (vel illuſtrati) puncto ad quodvis <lb/>medii (non obſtaculis interciſi) punctum lucis aliquis radius dirigitur. <lb/></s>
  <s xml:id="echoid-s376" xml:space="preserve">Hæc apud Opticos tritiſſima ſuppoſitio quò vel intelligi vel admitti <lb/>poſſit, omninò duplicem limitationem exigere videtur, è ſupra dictis <lb/>utramque deducibilem. </s>
  <s xml:id="echoid-s377" xml:space="preserve">Unam, ut omnis puncti nomine nedum non <lb/>præciſè punctum quodcunque Mathematicum, ac nec omnem particu-<lb/>lam concipiamus realem &amp; </s>
  <s xml:id="echoid-s378" xml:space="preserve">Phyſicam; </s>
  <s xml:id="echoid-s379" xml:space="preserve">verùm ſaltem admodum exi-<lb/>guam, qualíque ferme minorem vel animo deſignare nequeamus; </s>
  <s xml:id="echoid-s380" xml:space="preserve">al-<lb/>teram ut non in unoquoque ſtrictè dicto temporis inſtanti, nec in omni <lb/>reali temporis portiuncula cogitemus hoc contingere, ſed ut nullum <lb/>temporis intervallum ſentiri poſſit ità curtum, aut momentaneum, quin <lb/>intra ipſum à quavis lucidi deſignabili parte deſignatam ad medii partem <lb/>radius aliquis exporrigatur. </s>
  <s xml:id="echoid-s381" xml:space="preserve">Enimverò cùm radiorum iſtæ quas aſ-<lb/>ſignavimus radices, lucidum componentia corpuſcula, ſint illorum, <lb/>quorum nos utcunque quantitates ſenſu vel animo pertingere valemus, <lb/>corporum reſpectu tanquam infinitè parva, nec non infinità quaſi per-<lb/>nicitate donata, non difficilè concipi poteſt in omni deſignabili, vel <lb/>imaginabili lucentis ſpatiolo prorſus innumerabilem eorum multitudi-<lb/>tudinem exiſtere, quorum fere ſingula diverſas in plagas tendunt; </s>
  <s xml:id="echoid-s382" xml:space="preserve">ut <lb/>nulla ſit deſignabilis plaga, quam non una quæpiam appetat, aliquam <lb/>ſaltem, utlibèt imperceptibilis &amp; </s>
  <s xml:id="echoid-s383" xml:space="preserve">anguſti, temporis moram interpo-<lb/>nendo. </s>
  <s xml:id="echoid-s384" xml:space="preserve">In eo ſiquidem tempuſculo lucidi partes ſingulas innumera <lb/>ſucceſſivè talia corpuſcula ſubingrediuntur juxta deſerúntque, de qui-<lb/>bus mirum fuerit ni quoddam unum ad deſignatum medii ſpatium ten-<lb/>dat, ſibi tranſmittendo meatuum aliquem (quos &amp; </s>
  <s xml:id="echoid-s385" xml:space="preserve">pari ratione tan-
<pb o="8" file="0026" n="26" rhead=""/>
quam infinitos ſupponere fas eſt) idoneum reperiens. </s>
  <s xml:id="echoid-s386" xml:space="preserve">Ità vulgare pro-<lb/>nunciatum interpretor; </s>
  <s xml:id="echoid-s387" xml:space="preserve">id quod alias rigidè ſumptum haud verum du-<lb/>co. </s>
  <s xml:id="echoid-s388" xml:space="preserve">Nec enim idem corpus eodem temporis puncto diverſas in partes <lb/>contendere, vel adniti; </s>
  <s xml:id="echoid-s389" xml:space="preserve">ſed nec eandem præciſè medii partem è diver-<lb/>ſis locis accedentes corporum motus excipere quiſquam conceperit, <lb/>opinor, aut ego conceſſerim; </s>
  <s xml:id="echoid-s390" xml:space="preserve">non certè magìs quam idem corpus unà <lb/>plures locos occupare, vel eundem locum plura ſimul corpora ſuſcipe-<lb/>re; </s>
  <s xml:id="echoid-s391" xml:space="preserve">ad iſtum modum intellecta dicta ſuppoſitio totam unà cum radiis <lb/>lucidis naturam, omnem, utmihi videtur, Phyſicam permiſcebit. </s>
  <s xml:id="echoid-s392" xml:space="preserve">In <lb/>noſtro rem explicandi modo nihil durius obverſari video, quàm ut hinc <lb/>divinæ potentiæ, ſapientiæque vis magìs eluceſcat, in luce ſic efforman-<lb/>da, tam ejus effectricibus particulis admirabilem exilitatem, incom-<lb/>prehenſibilemque velocitatem impertiendo, quæ prorſus ei neceſſariæ <lb/>fuerunt, ut ſenſionem efficeret, &amp; </s>
  <s xml:id="echoid-s393" xml:space="preserve">reliqua tam utilia ei deſtinata mu-<lb/>nia obiret. </s>
  <s xml:id="echoid-s394" xml:space="preserve">Sanè lucis corpuſculum unum ab arenula quavis litorea <lb/>pluſquam eâ fortaſſis proportione ſuperatur, quâ tota quanta quanta <lb/>eſt mundana moles arenulam iſtam excedit; </s>
  <s xml:id="echoid-s395" xml:space="preserve">id quod non ita cenſe-<lb/>bit abſonum, quiſquis ad complures ſatìs obvias apparentias men-<lb/>tem adverterit.</s>
  <s xml:id="echoid-s396" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s397" xml:space="preserve">Subnotandum eſt porrò duas has fundamentales hypotheſes, ſic ac-<lb/>ceptas, innumeris adniodum familiaribus experimentis confirmari. <lb/></s>
  <s xml:id="echoid-s398" xml:space="preserve">Quovis enim in loco ubicunque collocati objecti lucentis vel illuſtrati <lb/>quæcunque deſignabilis particula conſpicitur oculo, repræſentatur in <lb/>ſpeculo, modò nihil objiciatur ab eo rectâ delabentes radios interclu-<lb/>dens; </s>
  <s xml:id="echoid-s399" xml:space="preserve">eadem verò ſtatim oculo ſubducitur, &amp; </s>
  <s xml:id="echoid-s400" xml:space="preserve">penitus obumbratur, <lb/>ſi quid opaci corporis directum intercipiens radiorum iter obtendatur. </s>
  <s xml:id="echoid-s401" xml:space="preserve"><lb/>Etiam foramen utcunque tantillum ſufficit trajiciendis radiis quibus tota <lb/>quantivis objecti facies obverſa depingatur. </s>
  <s xml:id="echoid-s402" xml:space="preserve">Et porrò quam nullà poſſit <lb/>apprehendi tam exigua lucidi pars, à qua non lux ad oculum defluit, <lb/>perſpiciliorum uſus apertiſſimè monſtrat. </s>
  <s xml:id="echoid-s403" xml:space="preserve">At pergo.</s>
  <s xml:id="echoid-s404" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s405" xml:space="preserve">X. </s>
  <s xml:id="echoid-s406" xml:space="preserve">3. </s>
  <s xml:id="echoid-s407" xml:space="preserve">Lucis radius quilibet alteri medio perpendiculariter incur-<lb/>rens, aut rectà progreditur, ſiquidem cedente medio procedere valet, <lb/>aut in partes directè contrarias (hoc eſt in ſe, vel in ſuam retro ſemi-<lb/>tam) repellitur. </s>
  <s xml:id="echoid-s408" xml:space="preserve">Experientiâ firmatur hæc hypotheſis; </s>
  <s xml:id="echoid-s409" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s410" xml:space="preserve">rationi <lb/>quoque conſentanea eſt; </s>
  <s xml:id="echoid-s411" xml:space="preserve">necenim ulla poteſt excogitari cauſa, cur in <lb/>unas potius quàm in alias partes deflectatur; </s>
  <s xml:id="echoid-s412" xml:space="preserve">igitur in nullas. </s>
  <s xml:id="echoid-s413" xml:space="preserve">Quinimô <lb/>ſiverum ſit omne patiens, aut percuſſum vim inferenti poſitivâ quâdam <lb/>vi repugnare, perſpicuum videtur eò reſiſtentiam dirigi, unde vis ingru-<lb/>ebat; </s>
  <s xml:id="echoid-s414" xml:space="preserve">ejúque conſequenter effectum abſolutè loquendo, tantùm illic
<pb o="9" file="0027" n="27" rhead=""/>
deprehendi. </s>
  <s xml:id="echoid-s415" xml:space="preserve">Quod ſanè mihi tam verum apparet, ut non dubitem <lb/>hancipſam hypotheſin ad omnimodos incurſus extendere; </s>
  <s xml:id="echoid-s416" xml:space="preserve">ſeu genera-<lb/>tim effari, quod pulſus omnis &amp; </s>
  <s xml:id="echoid-s417" xml:space="preserve">motus, utcunque medio culibet im-<lb/>pingens, directè (per ſe nimirum, propriè, diſtinctéque rem eſti-<lb/>mando) continuatur, aut prorſum aut retrorſum. </s>
  <s xml:id="echoid-s418" xml:space="preserve">Scilicet, exempli <lb/>cauſà ſi duo baculi A B Y Z, C D Y Z in idem medium E F (illud <lb/>perpendiculariter, hoc obliquè) uniformi quâdam preſſione vel impe-<lb/>tu adigantur, exiſtimo medii ceſſione vel reſiſtentiâ totam (quâ bacu-<lb/>lus obliquus fertur, aut medium impellit) vim æquè rectâ ſemità an-<lb/>trorſum verſus I K, vel retrò verſus C D derivari, ac perpendicularis <lb/>ipſius impetus in G H progreditur, aut regreditur in A B. </s>
  <s xml:id="echoid-s419" xml:space="preserve">Quod <lb/>enim nonnulli putant medii ſuperficiem baculi perpendicularis tenden-<lb/>tiæ magìs opponi, quam obliqui, proindéque perpendicularis impul-<lb/>ſum rectà continuari, ſed obliquum alio detorqueri; </s>
  <s xml:id="echoid-s420" xml:space="preserve">vel aſſertionem <lb/>ipſam non agnoſco, vel non admitto conſequentiam. </s>
  <s xml:id="echoid-s421" xml:space="preserve">Enimverò ſi per <lb/>illud opponi nil aliud volunt quàm realiter objici, ſeu obſtare recta <lb/>pergenti, non minùs eo modo ſuperſicies E F opponitur baculo C D, <lb/>
<anchor type="note" xlink:label="note-0027-01a" xlink:href="note-0027-01"/>
quàm ipſi A B; </s>
  <s xml:id="echoid-s422" xml:space="preserve">rectum enim ejus progreſſum pariter intercipit, im-<lb/>pedit, demutat. </s>
  <s xml:id="echoid-s423" xml:space="preserve">Verùm ſi quam aliam neſcio quam imaginariam op-<lb/>poſitionem intelligunt, nihil video quod huc faciat indè conſectari. <lb/></s>
  <s xml:id="echoid-s424" xml:space="preserve">Proſectò rem abſtractè, nec ut accidentarium quid immiſceamus, ex-<lb/>pendendo, nihil attinet ullam medii partem conſiderare præter illam, <lb/>ad quam corpus progrediens aut propellens ei occurrit; </s>
  <s xml:id="echoid-s425" xml:space="preserve">hæc enim ſola <lb/>reſiſtendo quicquam efficit, aut cedendo. </s>
  <s xml:id="echoid-s426" xml:space="preserve">Quare per rectam D Z pro-<lb/>gredienti impulſui ſolum punctum Z opponitur; </s>
  <s xml:id="echoid-s427" xml:space="preserve">perindéque fuerit <lb/>qualem reliqua medii ſuperficies obtinere ſitum concipiatur. </s>
  <s xml:id="echoid-s428" xml:space="preserve">Punctum <lb/>autem Z æquè pulſui venienti à D perrectam D Z, atque tendenti per <lb/>rectam Z K verſus K contrariatur, ac ei qui à B per B Z procedens iter <lb/>affectat per Z H verſus H. </s>
  <s xml:id="echoid-s429" xml:space="preserve">Idémque de reliquis medii punctis intelligi <lb/>par eſt, quibus uterque baculus ipſum contingit, aut ei applicatur. </s>
  <s xml:id="echoid-s430" xml:space="preserve"><lb/>Itaque reverà par utriuſque pulſùs quoad oppoſitionem eſt ratio; </s>
  <s xml:id="echoid-s431" xml:space="preserve">ſimi-<lb/>líſque proinde utrobique reſultabit effectus; </s>
  <s xml:id="echoid-s432" xml:space="preserve">pulſumnempe recto tra-<lb/>mite vel tranſmittere, vel rejicere. </s>
  <s xml:id="echoid-s433" xml:space="preserve">Verùm longè ſecus eveniet, ſi ba-<lb/>culum alterum obliquum, ſeu P D Y Q, cum ipſo A B Y Z confera-<lb/>mus Etenim ſuperſicies E F baculi A B Y Z motui, vel impulſui <lb/>magìs opponitur, aut obſiſtit, quàm motui vel impulſui baculi P D Y Q. </s>
  <s xml:id="echoid-s434" xml:space="preserve"><lb/>Quoniam illi toti cum tota ſui parte Y Z, huic vero tantum ex parte Y <lb/>renititur. </s>
  <s xml:id="echoid-s435" xml:space="preserve">è qua diſcrepantia neceſſariò diſpar effectus conſequetur, ut <lb/>nimirum pulſùs aut motûs directio mutetur. </s>
  <s xml:id="echoid-s436" xml:space="preserve">Quod diſcrimen eò lu-<lb/>bentius adnoto, quoniam hoc arbitror modo (vel adſimili) lucis ra-
<pb o="10" file="0028" n="28" rhead=""/>
dios diverſo medio obliquè incidentes, velut experimur, inflecti; </s>
  <s xml:id="echoid-s437" xml:space="preserve">ſal-<lb/>tem eò ſpectantia lucis præcipua ſymptomata, tribus porrò ſubjiciendis <lb/>hypotheſibus comprehenſa, vix aliâ ratione commodius explicari.</s>
  <s xml:id="echoid-s438" xml:space="preserve"/>
</p>
<div xml:id="echoid-div10" type="float" level="2" n="2">
<note position="right" xlink:label="note-0027-01" xlink:href="note-0027-01a" xml:space="preserve">Fig. 1.</note>
</div>
<p>
  <s xml:id="echoid-s439" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s440" xml:space="preserve">4. </s>
  <s xml:id="echoid-s441" xml:space="preserve">Omnis radii lucidi inflectio (hoc ſubinde generali nomine, <lb/>compendii cauſà, tam r@fractionem, quàm reflectionem complector) <lb/>fit in ſuperficie ad medii inflectentis ſuperficiem perpendiculari, ſeu <lb/>recta. </s>
  <s xml:id="echoid-s442" xml:space="preserve">Hujuſce ſuppoſitionis haud ullam facilè ſatìs commodam &amp; </s>
  <s xml:id="echoid-s443" xml:space="preserve"><lb/>claram rationem reperias apud Opticos; </s>
  <s xml:id="echoid-s444" xml:space="preserve">petitione principii, vel in-<lb/>comprehenſibili quâdam obſcuritate laborat quicquid fermè eò ſpectans <lb/>afferunt; </s>
  <s xml:id="echoid-s445" xml:space="preserve">neque valdè miror radium lucis ſemper ut rectam concipien-<lb/>tibus individuam lineam id eis accidiſſe; </s>
  <s xml:id="echoid-s446" xml:space="preserve">quo poſito vix probam ullam <lb/>ejuſce rei cauſam aſſignari poſſe credo. </s>
  <s xml:id="echoid-s447" xml:space="preserve">Cadat enim radius linearis <lb/>A B in ſpeculi (inſtantiæ gratiâ) plani ſuperficiem ad punctum B; </s>
  <s xml:id="echoid-s448" xml:space="preserve">per <lb/>
<anchor type="note" xlink:label="note-0028-01a" xlink:href="note-0028-01"/>
quod utcunque ducantur duæ rectæ C D, E F; </s>
  <s xml:id="echoid-s449" xml:space="preserve">cùm igitur rectæ A B, <lb/>C D ſint in uno quodam plano, quidni reflectio radii peragatur in iſto <lb/>plano? </s>
  <s xml:id="echoid-s450" xml:space="preserve">Simili ratione quoniam rectæ A B, E F ſunt in uno plano, <lb/>quidni radius in hoc etiam reflectionem patiatur? </s>
  <s xml:id="echoid-s451" xml:space="preserve">eodémque planè <lb/>modo quìd obſtat quo minùs in ſingulis omnibus, hoc eſt infinitis <lb/>planis, ſpeculi ſuperficiem ſecantibus, &amp; </s>
  <s xml:id="echoid-s452" xml:space="preserve">per rectam A B ceu com-<lb/>munem ſectionem traductis perficiatur reſlectio, idémque proinde ra-<lb/>dius unus in partes undique cunctas reflexus diſpergatur? </s>
  <s xml:id="echoid-s453" xml:space="preserve">cur hoc fieri <lb/>non poſſit, utique non capio. </s>
  <s xml:id="echoid-s454" xml:space="preserve">Quod reſpondetur enim, poſito plano <lb/>A B C ad ſpeculi ſuperficiem recto magis illud planum, quam cætera <lb/>quævis ſpeculi ſuperficiei contrarium eſſe, proindè reſiſtentiam in <lb/>eo maximam contingere, proptereáque radium in eo potiſſimùm in-<lb/>flecti, parùm ſatisfacit; </s>
  <s xml:id="echoid-s455" xml:space="preserve">quoniam, ut ſuperiùs inſinuatum, extra <lb/>punctum ipſum B, cui radius impingit, alia nulla ſpecularis ſuperficici <lb/>pars meritò venit conſideranda quid enim (ut hoc adjiciam prædictis) <lb/>an in univerſam quà longè latéque diſtenditur, ipſius ſpeculi ſuperfi-<lb/>ciem agit hic linearis radius, &amp; </s>
  <s xml:id="echoid-s456" xml:space="preserve">ab ea viciſſim patitur; </s>
  <s xml:id="echoid-s457" xml:space="preserve">an in ejus deſi-<lb/>nitam aliquam partem agit, patitúrque ab hac? </s>
  <s xml:id="echoid-s458" xml:space="preserve">quis in totam agere, <lb/>vel à tota pati concedet? </s>
  <s xml:id="echoid-s459" xml:space="preserve">Et cur id uni parti deputandum præ aliis? <lb/></s>
  <s xml:id="echoid-s460" xml:space="preserve">ubi terminus figetur? </s>
  <s xml:id="echoid-s461" xml:space="preserve">quouſque procedet operatio? </s>
  <s xml:id="echoid-s462" xml:space="preserve">quinimò potiùs, <lb/>quia radii per rectam A B procurrentis impulſui tantùm id ſpeculi quod <lb/>eſt in recta A B verſus G protracta reſiſtit, ideò pulſus in ipſam A B <lb/>rejicietur; </s>
  <s xml:id="echoid-s463" xml:space="preserve">Et nulla ſuccurrit cauſa ſontica, propter quam aliorſum <lb/>deflectat; </s>
  <s xml:id="echoid-s464" xml:space="preserve">nihil datur, quod ejus tendentiam aliò determinet. </s>
  <s xml:id="echoid-s465" xml:space="preserve">Igitur <lb/>ut aliis, quæ puto variæ aſſignantur, hujus effecti cauſis excutiendis <lb/>abſtineam, indè genuinam ejuſce rationem (ut &amp; </s>
  <s xml:id="echoid-s466" xml:space="preserve">generatim omnium
<pb o="11" file="0029" n="29" rhead=""/>
quæ circa radiorum inflectionem primitùs obveniunt) exiſtimo peten-<lb/>dam, quod lucis radius non mera ſit linea, verùm dimenſionibus om-<lb/>nimodis præditum corpus; </s>
  <s xml:id="echoid-s467" xml:space="preserve">utpote (juxta quæ præmonuimus) cylin-<lb/>dricum aut priſmaticum, pro figura corpuſculi, a quo oritur. </s>
  <s xml:id="echoid-s468" xml:space="preserve">Sup-<lb/>ponatur, aliquatenus illuſtrandi propoſiti ergò, Parallelepipedum <lb/>
<anchor type="note" xlink:label="note-0029-01a" xlink:href="note-0029-01"/>
ABCDEFGH lucis radium obliquè ſpeculo incurrentem repræ-<lb/>ſentare; </s>
  <s xml:id="echoid-s469" xml:space="preserve">cujus latus B F applicetur ſpeculo, dum interea reliquum <lb/>ejus ſupra ſpeculi planum elevatur. </s>
  <s xml:id="echoid-s470" xml:space="preserve">Impedietur ergò Parallelogramum <lb/>ABFE, nè recta procedat; </s>
  <s xml:id="echoid-s471" xml:space="preserve">indè continget rectam BF aliquò ſupra <lb/>dictum planum reſilire. </s>
  <s xml:id="echoid-s472" xml:space="preserve">Verùm in allas ſaltem partes fiet hæc refle-<lb/>ctio, ſecundum quas rectus radii progreſſus, quoad ejus fieri poteſt, <lb/>quàm minimè pervertetur. </s>
  <s xml:id="echoid-s473" xml:space="preserve">Cùm enim is rectiſſimum curſum affectet, <lb/>eum (ex indole certa, perpetuáque lege naturæ) ſi perfectè nequit, <lb/>at tamen ut proximè conſequetur. </s>
  <s xml:id="echoid-s474" xml:space="preserve">Itaque cùm inter plana latera <lb/>ABDC, EFHG ſibimet oppoſita curſus ejus anteà dirigeretur, &amp; </s>
  <s xml:id="echoid-s475" xml:space="preserve"><lb/>objecta ſuperficies nihil jam obſtet, quo minùs inter eadem plana, ta-<lb/>metſi ſurſum excuſſus, progrediatur, admodum liquet etiamnum inter <lb/>illa ſemitam ejus contineri; </s>
  <s xml:id="echoid-s476" xml:space="preserve">locumque ſeu plagam reflexionis eatenus <lb/>haud perperam determinari. </s>
  <s xml:id="echoid-s477" xml:space="preserve">Cæterùm eſt planum ABDC, eíque <lb/>oppoſitum EFGH ſpeculi plano rectum; </s>
  <s xml:id="echoid-s478" xml:space="preserve">quia Parallelepipedum <lb/>rectum ponitur, &amp; </s>
  <s xml:id="echoid-s479" xml:space="preserve">ideò lateralisrecta B F in ſpeculi plano exiſtens, <lb/>planis ABDC, EFHG recta. </s>
  <s xml:id="echoid-s480" xml:space="preserve">Quocircà ſitotum hoc Paralleledipedum <lb/>ob exilitatem ſuam, aut Mathematicæ computationis gratiâ, pro recta <lb/>quaſi linea cenſeatur, erit pariter &amp; </s>
  <s xml:id="echoid-s481" xml:space="preserve">reflexus radius etiam linea recta; <lb/></s>
  <s xml:id="echoid-s482" xml:space="preserve">nec non uterque continebitur in ſuperficie ad ſpeculi planum recta. </s>
  <s xml:id="echoid-s483" xml:space="preserve"><lb/>Non diſſimili ratiocinio, ſi radius cylindri recti figura præditus admit-<lb/>tatur (qualis nimirum à corpore procurrente, vel impulſo producetur, <lb/>id ſi Sphæricum fuerit) etiam radius in ſuperficie plano ſpeculi recta <lb/>reflectionem oſtendetur ſubire. </s>
  <s xml:id="echoid-s484" xml:space="preserve">Speculi quippe plano rectus incidat <lb/>cylindrus ABDC; </s>
  <s xml:id="echoid-s485" xml:space="preserve">cujus baſes AMCN, BODP, axis XZ; </s>
  <s xml:id="echoid-s486" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0029-02a" xlink:href="note-0029-02"/>
ità ſcilicet, ut baſis BODP ſpeculi planum contingat in B; </s>
  <s xml:id="echoid-s487" xml:space="preserve">reli-<lb/>quum ejus corpus (prout in figura depictum exhibetur) obliquè ſur-<lb/>gens ſupra planum emineat. </s>
  <s xml:id="echoid-s488" xml:space="preserve">Baſis autem diametri B D, P O ſeſe nor-<lb/>maliter ſecent; </s>
  <s xml:id="echoid-s489" xml:space="preserve">ac per ipſam P O, &amp; </s>
  <s xml:id="echoid-s490" xml:space="preserve">axem ductum planum efficiat in <lb/>cylindro Parallelogrammum P O M N. </s>
  <s xml:id="echoid-s491" xml:space="preserve">Si jam per hujuſce latera <lb/>MO, NP ducta concipiantur duo plana axi parallela, cylindrúmque <lb/>contingentia, liquebit (ex antedictis cauſis pariter applicatis) totius <lb/>cylindri ductum inter hæc duo plana comprehendi, radiique reflecti-<lb/>onem inter ipſa definiri. </s>
  <s xml:id="echoid-s492" xml:space="preserve">Sunt autem hæc plana ſpeculi plano recta. <lb/></s>
  <s xml:id="echoid-s493" xml:space="preserve">Sit enim recta G B H communis ſectio circnli B O D P, planíque
<pb o="12" file="0030" n="30" rhead=""/>
ſpecularis; </s>
  <s xml:id="echoid-s494" xml:space="preserve">hæc utique circulum continget; </s>
  <s xml:id="echoid-s495" xml:space="preserve">(quia ſpeculi planum, ex <lb/>hypotheſi, non alibi præterquam ad B circulo occurrit, adeóque nec <lb/>rec<unsure/>ta G H) quare rectæ G H, O P ſunt parallelæ. </s>
  <s xml:id="echoid-s496" xml:space="preserve">Ergo P O eſt ad <lb/>ſpeculi planum parallela. </s>
  <s xml:id="echoid-s497" xml:space="preserve">Huic verò perpendicularia ſunt plana præ-<lb/>dicta cylindrum contingentia per M O, N P ducta; </s>
  <s xml:id="echoid-s498" xml:space="preserve">axi parallela. <lb/></s>
  <s xml:id="echoid-s499" xml:space="preserve">Quapropter eadem ſpeculi plano recta erunt. </s>
  <s xml:id="echoid-s500" xml:space="preserve">Hinc, ut anteà ſi totus <lb/>radius habeatur inſtar rectæ lineæ, continget ejus reflectio velut in ſu-<lb/>perficie ad ſpeculum planum recta; </s>
  <s xml:id="echoid-s501" xml:space="preserve">quippe cùm ejus latitudo tota com-<lb/>prehendatur inter ejuſniodi duo plana; </s>
  <s xml:id="echoid-s502" xml:space="preserve">quæ proinde ſi nulla ſuppona-<lb/>tur, in unum illa coaleſcent. </s>
  <s xml:id="echoid-s503" xml:space="preserve">Accommodari poſſent hæc cuicunque <lb/>radii figuræ tali, qualem ſupra deſcripſimus, utcunque nonnulla de-<lb/>mutando; </s>
  <s xml:id="echoid-s504" xml:space="preserve">ſed &amp; </s>
  <s xml:id="echoid-s505" xml:space="preserve">eadem pari ratione radiorum refractionibus adapten-<lb/>tur. </s>
  <s xml:id="echoid-s506" xml:space="preserve">Atpluribus parco.</s>
  <s xml:id="echoid-s507" xml:space="preserve"/>
</p>
<div xml:id="echoid-div11" type="float" level="2" n="3">
<note position="left" xlink:label="note-0028-01" xlink:href="note-0028-01a" xml:space="preserve">Fig. 2.</note>
<note position="right" xlink:label="note-0029-01" xlink:href="note-0029-01a" xml:space="preserve">Fig. 3.</note>
<note position="right" xlink:label="note-0029-02" xlink:href="note-0029-02a" xml:space="preserve">Fig. 4.</note>
</div>
</div>
<div xml:id="echoid-div13" type="section" level="1" n="10">
<head xml:id="echoid-head13" xml:space="preserve"><emph style="sc">Lect.</emph> II.</head>
<p>
  <s xml:id="echoid-s508" xml:space="preserve">1. </s>
  <s xml:id="echoid-s509" xml:space="preserve">V Iæ, quam nuper aperuimus, &amp; </s>
  <s xml:id="echoid-s510" xml:space="preserve">aliquatenus ingreſſi ſumus, <lb/>inhærentes eò jam devenimus, ut nobis incumbat proximè <lb/>celebres illas hypotheſes (an Theoremata malitis appellare) radiorum <lb/>inflexorum itineri penitus determinando (imaginúmque proinde locis, <lb/>figuris, quantitatibus inveſtigandis, nec non apparentiarum quarum-<lb/>cunque cauſis explicandis) neceſſarias, experientiæ quidem bene con-<lb/>ſonas illas, etiam aliquo rationis ſuffragio communire; </s>
  <s xml:id="echoid-s511" xml:space="preserve">præſtratis <lb/>utique ſundamentis, ac ſuppoſitionibus inſiſtendo. </s>
  <s xml:id="echoid-s512" xml:space="preserve">Cùm itaque lucis <lb/>radio corpus adſignatum ſit figurâ priſmaticum, aut cylindricum; </s>
  <s xml:id="echoid-s513" xml:space="preserve">Et <lb/>hoc quidem rectum (utpote præ reliquis ſimplex, &amp; </s>
  <s xml:id="echoid-s514" xml:space="preserve">naturæ totas ſuas <lb/>in agendo vires exerenti præſertim conveniens;) </s>
  <s xml:id="echoid-s515" xml:space="preserve">cùm &amp; </s>
  <s xml:id="echoid-s516" xml:space="preserve">exinde pro-<lb/>greſſus ejus eatenus fuerit definitus, ut intra ſuperficies duas planas in-<lb/>Hectenti medio perpendiculares includatur; </s>
  <s xml:id="echoid-s517" xml:space="preserve">quas quidem abhinc (quan-<lb/>do nullus tranſverſæ dimenſionis illius, vel intervalli ſuperficies iſtas <lb/>dirimentis ad rem noſtram, illam ſaltem quam nunc attingimus ſpe-<lb/>ctans effectus, aut uſus ſit) brevitatis &amp; </s>
  <s xml:id="echoid-s518" xml:space="preserve">perſpicuitatis cauſâ, velut u-<lb/>nam habere poſſumus; </s>
  <s xml:id="echoid-s519" xml:space="preserve">adeóque jam radium ut duabus ſolummodò di-<lb/>menſionibus præditum, &amp; </s>
  <s xml:id="echoid-s520" xml:space="preserve">ad inſtar Parallelogrammi cujuſdam rect-<lb/>anguli, in plano ad medii inflectentis ſuperficiem recto jacentis, con-<lb/>fiderantes, reliquam itineris quod perſequitur determinationem, ulti-
<pb o="13" file="0031" n="31" rhead=""/>
mam illam &amp; </s>
  <s xml:id="echoid-s521" xml:space="preserve">completam, inveſtigabimus, ac exponemus; </s>
  <s xml:id="echoid-s522" xml:space="preserve">cujuſce <lb/>quidem circa reflectionem inquiſitionis conſectaria reſultabit hæc pro-<lb/>poſitio, paſſim ab Opticis recepta:</s>
  <s xml:id="echoid-s523" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s524" xml:space="preserve">II. </s>
  <s xml:id="echoid-s525" xml:space="preserve">5. </s>
  <s xml:id="echoid-s526" xml:space="preserve">_Radius inßidens, &amp; </s>
  <s xml:id="echoid-s527" xml:space="preserve">reflexus ad ſpeculi, velopaci reflectentis_ <lb/>_ſuperficiem angulos conſtituunt aquales_. </s>
  <s xml:id="echoid-s528" xml:space="preserve">Hujus effati declarationem <lb/>ſic exequimur. </s>
  <s xml:id="echoid-s529" xml:space="preserve">Parallelogramum rectangulum ABCD lucis repræ-<lb/>ſentet radium obliquè plano ſpeculo EF incidentem. </s>
  <s xml:id="echoid-s530" xml:space="preserve">(Recta ſcilicet <lb/>EF ſit communis ſectio plani ad ſpeculum re@ i, in quo dictum Paral-<lb/>lelogrammum exiſtit, &amp; </s>
  <s xml:id="echoid-s531" xml:space="preserve">in quo, ſecundum præmiſſa, reflectio per-<lb/>agitur, cum plano ſpeculi.) </s>
  <s xml:id="echoid-s532" xml:space="preserve">Cum itaque Parallelogrammi punctum B <lb/>ſpeculo primùm impingens opaco acimpervio, recta progredi nequeat, <lb/>conetur oportet (ut præſtruximus) retrò verſus A per ipſam rectam <lb/>BA reſilire. </s>
  <s xml:id="echoid-s533" xml:space="preserve">Cùm autem intereà rectæ BD ſupra ſpeculum eminen-<lb/>tis alter terminus D, nullo præpeditus obſtaculo pari vehementiâ cur-<lb/>ſum quoque ſuum adnitatur promovere per rectam CDH; </s>
  <s xml:id="echoid-s534" xml:space="preserve">palam <lb/>videtur utriuſque conatibus adverſis non aliter faciliùs aut propiùs ſa-<lb/>tisfieri poſſe, quàm ſi utrumque circa punctum Z rectæ BD medium <lb/>r@tationem concipiat. </s>
  <s xml:id="echoid-s535" xml:space="preserve">Sic enim utrumque pariter &amp; </s>
  <s xml:id="echoid-s536" xml:space="preserve">quàm minimum <lb/>à recto quem affectent curſu deflectent; </s>
  <s xml:id="echoid-s537" xml:space="preserve">ſiquidem rectæ BA, DC <lb/>circulum B β D δ tangunt, centro Z per B &amp; </s>
  <s xml:id="echoid-s538" xml:space="preserve">D deſcriptum. </s>
  <s xml:id="echoid-s539" xml:space="preserve">Cùm <lb/>autem hujuſmodi motum circularem obeundo punctum B deſcripſerit <lb/>arcum B β, &amp; </s>
  <s xml:id="echoid-s540" xml:space="preserve">punctum D arcum D δ, hoc eſt quando recta BD ob-<lb/>tinuerit ſitum β δ, etiam ipſum punctum D ſpeculo impinget ad δ; <lb/></s>
  <s xml:id="echoid-s541" xml:space="preserve">reditúmque proinde per arcum δ D, ſcilicet ipſius quoque jam inter-<lb/>ciſo curſu, molietur; </s>
  <s xml:id="echoid-s542" xml:space="preserve">Sed &amp; </s>
  <s xml:id="echoid-s543" xml:space="preserve">nunc temporis ipſum punctum B ad β po-<lb/>ſitum per arcum β D tendit; </s>
  <s xml:id="echoid-s544" xml:space="preserve">quorum certè motuum adverſantium al-<lb/>ter alterius effectum impediet; </s>
  <s xml:id="echoid-s545" xml:space="preserve">itáque proximo ſaltem, quoad fieri <lb/>poterit, utrumque progreſſus arripient; </s>
  <s xml:id="echoid-s546" xml:space="preserve">proximi vero ſunt qui per <lb/>tangentes β α, δ κ; </s>
  <s xml:id="echoid-s547" xml:space="preserve">qui &amp; </s>
  <s xml:id="echoid-s548" xml:space="preserve">ſibi nihil repugnant, at potiùs omninò ſe-<lb/>cum conſpirant; </s>
  <s xml:id="echoid-s549" xml:space="preserve">itaque punctum B per rectam β κ, punctúmque D per <lb/>rectam β κ procurrent, adeò ut totus radius ABDC jam acquirat <lb/>ſitum α β δ κ; </s>
  <s xml:id="echoid-s550" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s551" xml:space="preserve">per hanc orbitam recta motum ſuum proſequatur. </s>
  <s xml:id="echoid-s552" xml:space="preserve"><lb/>Liquet autem angulos ABF, κ δE æquari. </s>
  <s xml:id="echoid-s553" xml:space="preserve">Nam æquantur anguli <lb/>ZB δ, Z δ B; </s>
  <s xml:id="echoid-s554" xml:space="preserve">quapropter adjunctis hinc indè rectis ZBA, β δ κ toti <lb/>ABF, κ δ E pares erunt. </s>
  <s xml:id="echoid-s555" xml:space="preserve">Unde patet è duobus quoque rectis reſiduos <lb/>
<anchor type="note" xlink:label="note-0031-01a" xlink:href="note-0031-01"/>
ABE, κ δ F æquari; </s>
  <s xml:id="echoid-s556" xml:space="preserve">quod propoſitum fuit oſtendere.</s>
  <s xml:id="echoid-s557" xml:space="preserve"/>
</p>
<div xml:id="echoid-div13" type="float" level="2" n="1">
<note position="right" xlink:label="note-0031-01" xlink:href="note-0031-01a" xml:space="preserve">Fig. 5.</note>
</div>
<p>
  <s xml:id="echoid-s558" xml:space="preserve">III. </s>
  <s xml:id="echoid-s559" xml:space="preserve">Ità de præmiſſis ſuppoſitionibus noſtris fundamentalem hanc <lb/>Caεθptricæ legem ſeu regulani elicimus, quàm veriſimiliter aut con-
<pb o="14" file="0032" n="32" rhead=""/>
cinnè penes vos eſto judicium. </s>
  <s xml:id="echoid-s560" xml:space="preserve">Non diffitebor autem aut penitus diſſi-<lb/>mulabo non eſſe nihil quod his objici poſſit, &amp; </s>
  <s xml:id="echoid-s561" xml:space="preserve">dubitandi cauſam inji-<lb/>cere. </s>
  <s xml:id="echoid-s562" xml:space="preserve">Cur enim, percontetur aliquis, quando ſolum punctum B vet-<lb/>ſus A renitatur, &amp; </s>
  <s xml:id="echoid-s563" xml:space="preserve">totum lineæ BD quod ſupereſt partes appetat <lb/>contrarias, non circa punctum quodpiam aliud in ipſa BD, puncto B <lb/>propinquius, ut puta circa X, potiùs iſta gyratio concipiatur peragen-<lb/>da? </s>
  <s xml:id="echoid-s564" xml:space="preserve">Reſpondeo quàm breviſſimè (quonia@ incitato curſu tendens ul-<lb/>
<anchor type="note" xlink:label="note-0032-01a" xlink:href="note-0032-01"/>
teriùs æ grè remoras fert) id in natura conſtanter accidere, quum mo-<lb/>tus rectus in circularem degenerat, ut extremæ ſibimet adverſæ mobi-<lb/>lium partes omnem motum dirigant ac moderentur, reliquis ad illa-<lb/>rum ductum componentibus ſe, motúſque ſuos attemperantibus; </s>
  <s xml:id="echoid-s565" xml:space="preserve">ne-<lb/>que non his quos ob extremarum contraniſmum, atque conflictum <lb/>amittere neceſſe habent in illas transfundentibus; </s>
  <s xml:id="echoid-s566" xml:space="preserve">quo fit ut mediis <lb/>hinc indè quàm tardiſſimè dimotis extremæ velociùs revolvantur, Ita-<lb/>que cùm extrema puncta B, D partes in contrarium æquâ vi nitantur, <lb/>neque niſi circa medium punctum Z rotatio peragatur, quod effectant <lb/>aſſequi poſſint, id ſtatim fiet, &amp; </s>
  <s xml:id="echoid-s567" xml:space="preserve">reliquæ partes haud gravatim obſe-<lb/>quentur. </s>
  <s xml:id="echoid-s568" xml:space="preserve">Nè dicam in recta BD nullum aliud punctum exiſtere, cui <lb/>præ aliis jure prærogativa competit, ut circa ipſum mobile libretur. <lb/></s>
  <s xml:id="echoid-s569" xml:space="preserve">At pluribus abſtinens ad refractionis præcipuam legem haud abſimili <lb/>diſcurſu proliciendam atque declarandam accedo. </s>
  <s xml:id="echoid-s570" xml:space="preserve">Hanc nempe:</s>
  <s xml:id="echoid-s571" xml:space="preserve"/>
</p>
<div xml:id="echoid-div14" type="float" level="2" n="2">
<note position="left" xlink:label="note-0032-01" xlink:href="note-0032-01a" xml:space="preserve">Fig. 6.</note>
</div>
<p>
  <s xml:id="echoid-s572" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s573" xml:space="preserve">6. </s>
  <s xml:id="echoid-s574" xml:space="preserve">Radii lucis alteri cuipiam diſſimili perſpicuo (nimirum ho-<lb/>mogeneo quoad ſe) incidentes ità refringuntur, ut perpetuò recti ſi-<lb/>nus inclinationum, quas habent incidentes, proportionales ſint rectis <lb/>ſinubus inclinationum, quas obtinent refracti. </s>
  <s xml:id="echoid-s575" xml:space="preserve">Huic elucidando, ſta-<lb/>biliendóque decreto; </s>
  <s xml:id="echoid-s576" xml:space="preserve">Parallelogrammum ABDC lucis radium re-<lb/>præſentans impingat planæ ſuperficiei EF pellucidi medii (vel ſit recta <lb/>EF ſectio communis, ut in caſu præcedente, quod &amp; </s>
  <s xml:id="echoid-s577" xml:space="preserve">abhinc ſemper <lb/>intelligatur) progreſſum ejus aliquatenus retundentis. </s>
  <s xml:id="echoid-s578" xml:space="preserve">Itaque medium <lb/>iſthoc ſubingrediens punctum B procedere, tardiùs quidem, attenta-<lb/>bit per rectam BG, ſeu peripſam AB protractam, intereà verò pun-<lb/>ctum D in primo durans medio motum ſuum priorem adurgebit in re-<lb/>cta CDH. </s>
  <s xml:id="echoid-s579" xml:space="preserve">Hos autem conatus, aliàs irritos futuros (nec enim utrum-<lb/>que poteſt rectum motum illud tardiùs, hoc velociùs incedendo conſer-<lb/>vare) quàm proximè conſequentur, modò circa punctum aliquod in <lb/>recta DB producta ſitum, puta quale Z, rotentur; </s>
  <s xml:id="echoid-s580" xml:space="preserve">ità ſcilicet ut dum <lb/>punctum D in medio rariori (rarius appello quod minùs reſiſtit, aut <lb/>retardat; </s>
  <s xml:id="echoid-s581" xml:space="preserve">ut &amp; </s>
  <s xml:id="echoid-s582" xml:space="preserve">denſius quod motum magis reprimit, &amp; </s>
  <s xml:id="echoid-s583" xml:space="preserve">tardiorem <lb/>reddit) velociùs latum deſcribit arcum majorem D δ; </s>
  <s xml:id="echoid-s584" xml:space="preserve">punctum B
<pb o="15" file="0033" n="33" rhead=""/>
tardiùs in medio contumaciore delatum minorem arcum B β delineet; <lb/></s>
  <s xml:id="echoid-s585" xml:space="preserve">quibus peractis recta BD tenebit ſitum β δ. </s>
  <s xml:id="echoid-s586" xml:space="preserve">Cùm verò jam punctum <lb/>D denſius quoque medium interet ad δ; </s>
  <s xml:id="echoid-s587" xml:space="preserve">proindéque pariter &amp; </s>
  <s xml:id="echoid-s588" xml:space="preserve">ipſum <lb/>retardetur; </s>
  <s xml:id="echoid-s589" xml:space="preserve">motus iſti circulares protinus extinguantur oportet (nec <lb/>enim jam punctum D velociùs feretur quàm B; </s>
  <s xml:id="echoid-s590" xml:space="preserve">nec ideò majorem ut <lb/>priùs ſimul arcum deſcribet.) </s>
  <s xml:id="echoid-s591" xml:space="preserve">Itaque prius iter, quàm poterunt proxi-<lb/>mè, deſerentia tendent utrumque per horum arcuum tangentes δ κ, <lb/>β α; </s>
  <s xml:id="echoid-s592" xml:space="preserve">radiúſque totus ABCD hoc modo detortus, &amp; </s>
  <s xml:id="echoid-s593" xml:space="preserve">ſitum α β δ κ <lb/>nactus per hanc poſteà ſemitam rectà decnrret Adnotandum eſt au-<lb/>tem quæcunque ſit rectæ AB ad rectam EF inclinatio arcus D δ, B β <lb/>(vel ſemidiametros ZD, ZB) eandem ſemper habere proportionem <lb/>inter ſe; </s>
  <s xml:id="echoid-s594" xml:space="preserve">talem nempe, qualem in denſitate, ſeu reſiſtentia peculiare <lb/>diſcrimen exigit. </s>
  <s xml:id="echoid-s595" xml:space="preserve">Etenim ſupponatur in quovis ſuperficiei pellucidæ <lb/>
<anchor type="note" xlink:label="note-0033-01a" xlink:href="note-0033-01"/>
loco poſitum nobile punctum B; </s>
  <s xml:id="echoid-s596" xml:space="preserve">cùm medium hoc ex hypotheſi ſit <lb/>homogeneum (hoc eſt ubique pariter obſiſtens) nulla poteſt, opinor <lb/>aſſignari ratio cur hoc mobile non in quaſvis partes æ quâ velocitate de-<lb/>ferri poſſit; </s>
  <s xml:id="echoid-s597" xml:space="preserve">nimirum æquè celeriter ad Q tendet, (impetum modò <lb/>ceperit iſthàc dirigentem) per rectam OBQ, ac in N per rectam <lb/>ABN. </s>
  <s xml:id="echoid-s598" xml:space="preserve">Adeóque radii lucidi AB, OB utcunque differenter inclinati <lb/>
<anchor type="note" xlink:label="note-0033-02a" xlink:href="note-0033-02"/>
parem omnino reſiſtentiam invenient; </s>
  <s xml:id="echoid-s599" xml:space="preserve">punctum, inquam, B, ſeu verſus <lb/>Q, ſeu verſus N nitatur, æqualiter, eodémque modo retardabitur. <lb/></s>
  <s xml:id="echoid-s600" xml:space="preserve">Quinetiam cùm punctum D in primo medio ſemper eâdem, quæcunque <lb/>fuerit ejus poſitio, celeritate promoveatur, ſatis apparet motus iſtos, <lb/>aut motuum ſemitas eodem tempore decurſas, arcus nempe circulares <lb/>D δ, B β ſemper eandem inter ſe proportionem ſervare; </s>
  <s xml:id="echoid-s601" xml:space="preserve">nimirum il-<lb/>lam, quam habent ſemidiametri ZD, ZB, vel Z δ, ZB; </s>
  <s xml:id="echoid-s602" xml:space="preserve">quæ idcir-<lb/>co proportio, principaliter ac primariò, radiorum refractiones, ad <lb/>eadem duo media factas, determinat atque metitur. </s>
  <s xml:id="echoid-s603" xml:space="preserve">Hanc autem ean-<lb/>dem eſſe patet cum illa, quam habent recti ſinus angulorum ipſis Zδ, <lb/>ZB in triangulo Z δ B oppoſitorum, ipſorum ſcilicet ZB δ (vel <lb/>ZBE) &amp; </s>
  <s xml:id="echoid-s604" xml:space="preserve">Z δ B. </s>
  <s xml:id="echoid-s605" xml:space="preserve">Eſt autem angulus ZBE complementum anguli <lb/>ABE, (hoc eſt angulus inclinationis rectæ AB ad EF) &amp; </s>
  <s xml:id="echoid-s606" xml:space="preserve">angulus <lb/>Z δ B eſt complementum anguli F δ κ, vel inclinatio rectæ δ κ ad ean-<lb/>dem EF. </s>
  <s xml:id="echoid-s607" xml:space="preserve">Igitur abunde liquet propoſitum. </s>
  <s xml:id="echoid-s608" xml:space="preserve">Patet vero, quod in hoc <lb/>caſu, angulus EBZ major eſt angulo B δ Z; </s>
  <s xml:id="echoid-s609" xml:space="preserve">vel, ductis BM, δ N <lb/>ad EF perpendicularibus, quòd angulus MBG major eſt angulo <lb/>N δ κ; </s>
  <s xml:id="echoid-s610" xml:space="preserve">adeóque quòd hic refractio verſus perpendicularem, quod ai-<lb/>unt. </s>
  <s xml:id="echoid-s611" xml:space="preserve">contingit. </s>
  <s xml:id="echoid-s612" xml:space="preserve">Ac ità quidem quando radius radius in medium tranſit, <lb/>ipſi magis obſiſtens, ſen denſiùs. </s>
  <s xml:id="echoid-s613" xml:space="preserve">At ſi medio incurrit faciliorem tran-<lb/>ſitum præbenti, ſeu rariori, planè ſimili modo, ſed inverſè ſe res habet.</s>
  <s xml:id="echoid-s614" xml:space="preserve">
<pb o="16" file="0034" n="34" rhead=""/>
Quod (licèt breviùs) conficenetur negotium adſumendo ſicut eadem <lb/>_Tbebis Atbenas, ac Atbenis Tbebas_ eſt via, ità radium de raro tranſe-<lb/>untem in denſius, pérque denſius veſtigia ſua replicantem in rarum <lb/>nil aliud quàm eandem ſemitam repetere; </s>
  <s xml:id="echoid-s615" xml:space="preserve">ut nempe ſi radius ABDC <lb/>de raro tranſiens in denſius refringatur in α β κ δ; </s>
  <s xml:id="echoid-s616" xml:space="preserve">quòd etiam hic ra-<lb/>dius α β κ δ è denſiori recidens in rarum viciſſimin ABDC refringe-<lb/>tur; </s>
  <s xml:id="echoid-s617" xml:space="preserve">quia tamen aſſumptum illud non nemini demonſtrationis &amp; </s>
  <s xml:id="echoid-s618" xml:space="preserve">ip-<lb/>ſum indigere videatur; </s>
  <s xml:id="echoid-s619" xml:space="preserve">Et univerſim, extremóque rigore ſumptum for-<lb/>ſan haud adeò verum ſit; </s>
  <s xml:id="echoid-s620" xml:space="preserve">majoris etiam evidentiæ cauſa; </s>
  <s xml:id="echoid-s621" xml:space="preserve">preſertîmq; <lb/></s>
  <s xml:id="echoid-s622" xml:space="preserve">demùm quoniam huic caſui nonnulla quodammodò peculiaria ſunt no-<lb/>tatu non indigna; </s>
  <s xml:id="echoid-s623" xml:space="preserve">quin addo quia præſtare videtur effectum unum-<lb/>quemque propriis è cauſis deduci) ſeparatim oſtendemus. </s>
  <s xml:id="echoid-s624" xml:space="preserve">Rurſum <lb/>igitur radius ABDC, quâ priùs figurâ donatus rarioris medii ſuper-<lb/>ficiem EF incurrat. </s>
  <s xml:id="echoid-s625" xml:space="preserve">Cùm igitur punctum B velociùs procedere jam va-<lb/>leat quàm anteà (medio ſcilicet illapſum promptiùs cedenti) hoc eſt <lb/>quàm punctum D, neceſſariò commutabitur rectus utriuſque, quem <lb/>affectant, motus in ei proximum circularem, circa punctum aliquod <lb/>in recta BD, puta circa Z; </s>
  <s xml:id="echoid-s626" xml:space="preserve">itâ ut ZD, ZB talem inter ſe proportio-<lb/>
<anchor type="note" xlink:label="note-0034-01a" xlink:href="note-0034-01"/>
nem obſervent, qualem ſingularis exigit horum in reſiſtentia mediorum <lb/>diverſitas; </s>
  <s xml:id="echoid-s627" xml:space="preserve">utique ſicut in quæ præceſſerunt; </s>
  <s xml:id="echoid-s628" xml:space="preserve">cùm verò punctum B <lb/>ità circumductum deſcripſerit arcum B β, &amp; </s>
  <s xml:id="echoid-s629" xml:space="preserve">punctum D arcum D δ; <lb/></s>
  <s xml:id="echoid-s630" xml:space="preserve">puncto D ad δ tunc medium rarius ingredienti, ceſſabit iſta motuum <lb/>inæqualitas; </s>
  <s xml:id="echoid-s631" xml:space="preserve">adeóque ſimul neceſſariò deſinet rotatus circa punctum <lb/>Z; </s>
  <s xml:id="echoid-s632" xml:space="preserve">ambóque puncta B, D per dictorum arcuum tangentes β α, δ κ (re-<lb/>ctæ Z β perpendiculares) quod proximum eſt iter arripient. </s>
  <s xml:id="echoid-s633" xml:space="preserve">Rurſus <lb/>autem, pariter ac in caſu præ cedente, rectæ ZD, ZB (vel Z δ, ZB) <lb/>proportionem exhibent, quæ refractiones hujuſmodi dimetitur; </s>
  <s xml:id="echoid-s634" xml:space="preserve">ha-<lb/>bent autem Z δ, ZB ſeipſas, ut recti ſinus angulorum ZB δ, Z δ E; </s>
  <s xml:id="echoid-s635" xml:space="preserve"><lb/>hoc eſt ut ſinus inclinationis rectæ AB ad ſinum inclinationis rectæ δ κ; </s>
  <s xml:id="echoid-s636" xml:space="preserve"><lb/>quod propoſitum fuit oſtendere. </s>
  <s xml:id="echoid-s637" xml:space="preserve">Liquet autem quòd hîc ang. </s>
  <s xml:id="echoid-s638" xml:space="preserve">Z δ F <lb/>major eſt angulo ZB δ, adeóque quod refractus divergit à per-<lb/>pendiculari.</s>
  <s xml:id="echoid-s639" xml:space="preserve"/>
</p>
<div xml:id="echoid-div15" type="float" level="2" n="3">
<note position="right" xlink:label="note-0033-01" xlink:href="note-0033-01a" xml:space="preserve">Fig. 7.</note>
<note position="right" xlink:label="note-0033-02" xlink:href="note-0033-02a" xml:space="preserve">Fig. 8.</note>
<note position="left" xlink:label="note-0034-01" xlink:href="note-0034-01a" xml:space="preserve">Fig. 9.</note>
</div>
<p>
  <s xml:id="echoid-s640" xml:space="preserve">V. </s>
  <s xml:id="echoid-s641" xml:space="preserve">Advertendum eſt porrò quoad priorem hypotheſin, ſeu caſum <lb/>radii de medio rariori contendentis in denſius, eum femper, qualiſcunq; <lb/></s>
  <s xml:id="echoid-s642" xml:space="preserve">fit ejus obliquitas, medium denſius ſubire; </s>
  <s xml:id="echoid-s643" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s644" xml:space="preserve">per ipſum incedere; </s>
  <s xml:id="echoid-s645" xml:space="preserve"><lb/>modo commonſtratotrato. </s>
  <s xml:id="echoid-s646" xml:space="preserve">[Simpliciter autem hoc, &amp; </s>
  <s xml:id="echoid-s647" xml:space="preserve">abſtractè debet in-<lb/>telligi, necut accidentarium quicquam interveniat, qualia ſunt, opaci-<lb/>tas perſpicuitati immiſta, figura diaphanum terminans, ejus craſſities <lb/>inæqualis, aliud quid poſt poſitum diaphani reſiſtentiam promovens;</s>
  <s xml:id="echoid-s648" xml:space="preserve">
<pb o="17" file="0035" n="35" rhead=""/>
cujuſmodi quippe de cauſis diaphanum ſubinde forſan evaſurum eſt <lb/>opacum, &amp; </s>
  <s xml:id="echoid-s649" xml:space="preserve">inſtar opaciradios valebit repercútere; </s>
  <s xml:id="echoid-s650" xml:space="preserve">ceu quando lapis <lb/>in aquam impingit obliquè; </s>
  <s xml:id="echoid-s651" xml:space="preserve">cùm hydrargyro ſubſtracto vitrum mu-<lb/>nitur. </s>
  <s xml:id="echoid-s652" xml:space="preserve">Dum lapis _e. </s>
  <s xml:id="echoid-s653" xml:space="preserve">g._ </s>
  <s xml:id="echoid-s654" xml:space="preserve">obliquè impingit ſuperficiei EF (cui parallela <lb/>OQ) per lineam AB; </s>
  <s xml:id="echoid-s655" xml:space="preserve">tota linea BQ ad fundum OQ protenſa <lb/>venienti repugnabit, auxilii quoque nonnihil conferente fundo OQ; <lb/></s>
  <s xml:id="echoid-s656" xml:space="preserve">neque mirum fuerit, ſi major hic renitentia deprehendatur, quam ubi <lb/>radius alter MBP perpendiculariús incurrit, quando major ſit BQ, <lb/>quam BP.</s>
  <s xml:id="echoid-s657" xml:space="preserve">] At nos ſecluſis iſtis medium velut interminatum, in om-<lb/>nes partes æqualiter reſiſtens, abſolutè perſpicuum, &amp; </s>
  <s xml:id="echoid-s658" xml:space="preserve">radios ex ſe <lb/>non reſpuens accipimus; </s>
  <s xml:id="echoid-s659" xml:space="preserve">quibus ſuppoſitis perpetuò, quod dixi, ra-<lb/>dius, obliquitate quâpiam incidentiæ nil vetante, medium denſius pe-<lb/>netrabit. </s>
  <s xml:id="echoid-s660" xml:space="preserve">Verum in alterò caſu, cùm de medio denſiori lux rarius in-<lb/>currit, non ſemper ea medium hoc permeabit. </s>
  <s xml:id="echoid-s661" xml:space="preserve">Nam ſi magna ſatìs <lb/>fuerit obliquitas; </s>
  <s xml:id="echoid-s662" xml:space="preserve">ſubinde radius inflexus ſupra ſuperficiem EF at-<lb/>tolletur, angulúſque (qui dicitur) refractus, aut inflexus rectum ex-<lb/>ſuperabit; </s>
  <s xml:id="echoid-s663" xml:space="preserve">quinimò fieri poteſt ut ipſum exæquet. </s>
  <s xml:id="echoid-s664" xml:space="preserve">Sit in exemplum <lb/>primò inclinatio graduum 45, vel ſemirecta; </s>
  <s xml:id="echoid-s665" xml:space="preserve">Et ZB ad ZD ſe habe-<lb/>at ut quadrati diameter ad ſuum latus quæ fermè proportio radiorum <lb/>ex aqua in aerem tranſeuntium, experientiâ conteſtante, rationem <lb/>
<anchor type="note" xlink:label="note-0035-01a" xlink:href="note-0035-01"/>
metitur) radius velut in ipſam EF refringetur, aut eam ſtringens pro-<lb/>cedet. </s>
  <s xml:id="echoid-s666" xml:space="preserve">Eſt enim Z δ (æqualis ipſi ZD) jam ad EF perpendicularis, <lb/>adeoque δ κ arcum δ D contingens ipſi EF congruet. </s>
  <s xml:id="echoid-s667" xml:space="preserve">Unde patet <lb/>obiter, id quod ſuperiùs inſinuatum, non univerſim conſtare, quòd <lb/>radius à quo loco medii unius in aliud proceſſit, ad eundem retrogra-<lb/>dus accedet. </s>
  <s xml:id="echoid-s668" xml:space="preserve">Hoc enim ſaltem in caſu radius AB refringitur in β α ſu-<lb/>perficiei media dirimenti parallelam; </s>
  <s xml:id="echoid-s669" xml:space="preserve">veruntamen qui per α β progre-<lb/>ditur minjmè recedet ad BA, nec ullam, ut manifeſtum eſt, omninò <lb/>refractionem patietur. </s>
  <s xml:id="echoid-s670" xml:space="preserve">Sed hic caſus tantùm unus, &amp; </s>
  <s xml:id="echoid-s671" xml:space="preserve">quaſi pro nullo <lb/>cenſeri poteſt. </s>
  <s xml:id="echoid-s672" xml:space="preserve">Quod ſi, ſervatâ quoad denſitatem eâdem proportione, <lb/>
<anchor type="note" xlink:label="note-0035-02a" xlink:href="note-0035-02"/>
radius AB paullò magis ad rectam EF inclinetur, ejus. </s>
  <s xml:id="echoid-s673" xml:space="preserve">Refractus <lb/>ſupra ipſam EF aſſurget; </s>
  <s xml:id="echoid-s674" xml:space="preserve">punctum quippe D rectam EF nunquam <lb/>pertinget; </s>
  <s xml:id="echoid-s675" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s676" xml:space="preserve">punctum B decurſâ rarius intra medium peripheriâ <lb/>B β in denſum remeabit; </s>
  <s xml:id="echoid-s677" xml:space="preserve">in quo proinde rurſus, circulatione ſuâ di-<lb/>miſſà, per tangentes β α, δ κ ferentur; </s>
  <s xml:id="echoid-s678" xml:space="preserve">adeò quidem ut radius ABCD <lb/>jam reflecti videatur, quatenus medium denſius haud penetrat totus, <lb/>vel egreditur.</s>
  <s xml:id="echoid-s679" xml:space="preserve"/>
</p>
<div xml:id="echoid-div16" type="float" level="2" n="4">
<note position="right" xlink:label="note-0035-01" xlink:href="note-0035-01a" xml:space="preserve">Fig. 12.</note>
<note position="right" xlink:label="note-0035-02" xlink:href="note-0035-02a" xml:space="preserve">Fig. 13.</note>
</div>
<p>
  <s xml:id="echoid-s680" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s681" xml:space="preserve">Nec ineptè quidem (etſi quodammodò, velutíque primariò, ſit <lb/>refractio) reflectionis nomen adſciſcit hæc actio, quatenus &amp; </s>
  <s xml:id="echoid-s682" xml:space="preserve">ipſa
<pb o="18" file="0036" n="36" rhead=""/>
reflectionis leges examuſſim obſervat. </s>
  <s xml:id="echoid-s683" xml:space="preserve">Nam quoniam iſoſcelis trian-<lb/>guli ZB β anguli ZB β Z β B ſunt æquales, etiam anguli (de rectis <lb/>reſidui) ABE, α β F pares erunt; </s>
  <s xml:id="echoid-s684" xml:space="preserve">quod reflectioni proprium eſt. <lb/></s>
  <s xml:id="echoid-s685" xml:space="preserve">Itaque non abs recto pronunciant hoc Dioptrici; </s>
  <s xml:id="echoid-s686" xml:space="preserve">neque tamen cauſam, <lb/>
<anchor type="note" xlink:label="note-0036-01a" xlink:href="note-0036-01"/>
fortaſſis ab iis prætermiſſam, tacere volui, nonnihil ab immediatæ re-<lb/>flectionis cauſa diverſam; </s>
  <s xml:id="echoid-s687" xml:space="preserve">nè quiſquam hæſitet, aut hoc adſumenti <lb/>gravetur concedere.</s>
  <s xml:id="echoid-s688" xml:space="preserve"/>
</p>
<div xml:id="echoid-div17" type="float" level="2" n="5">
<note position="left" xlink:label="note-0036-01" xlink:href="note-0036-01a" xml:space="preserve">_Kepl. prop. 14_.</note>
</div>
<p>
  <s xml:id="echoid-s689" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s690" xml:space="preserve">Hæc autem doctrina cùm multis experimentis utcunque com-<lb/>probari queat, unum ſaltem breviter attingam ſatìs illuſtre, neminíſq; <lb/></s>
  <s xml:id="echoid-s691" xml:space="preserve">non examini patens. </s>
  <s xml:id="echoid-s692" xml:space="preserve">Eſto triangulum ABC ſectio priſmatis triangu-<lb/>laris æquilateri (nimirum vitrei, ſeu cryſtallini) baſi parallela; </s>
  <s xml:id="echoid-s693" xml:space="preserve">in hu-<lb/>
<anchor type="note" xlink:label="note-0036-02a" xlink:href="note-0036-02"/>
jus autem baſe ſumatur punctum quodpiam F, &amp; </s>
  <s xml:id="echoid-s694" xml:space="preserve">ſit angulus CFG <lb/>circiter graduum 50. </s>
  <s xml:id="echoid-s695" xml:space="preserve">(unde juxta doctrinam hîc inſinuatam, &amp; </s>
  <s xml:id="echoid-s696" xml:space="preserve">poſteà <lb/>clariùs exprimendam, radius GF velut extremus erit eorum, qui re-<lb/>ctæ BC è vitri partibus illabentes refractionem patientur; </s>
  <s xml:id="echoid-s697" xml:space="preserve">eo ſcilicet <lb/>obliquior quilibet reflectetur.) </s>
  <s xml:id="echoid-s698" xml:space="preserve">Sit igitur (quoniam utramque quaſi <lb/>patitur inflectionem) ejus refractus FM, reflexus FE; </s>
  <s xml:id="echoid-s699" xml:space="preserve">item FGre-<lb/>fringatur in GO, &amp; </s>
  <s xml:id="echoid-s700" xml:space="preserve">FE in ER. </s>
  <s xml:id="echoid-s701" xml:space="preserve">Porrò jam in GO ſtatuatur oculi <lb/>centrum O; </s>
  <s xml:id="echoid-s702" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s703" xml:space="preserve">ab eo prodeat radius OQ, qui refringatur in QP; <lb/></s>
  <s xml:id="echoid-s704" xml:space="preserve">ipſorum OG, OQ refracti GF, OP (uti ſecundum principia no-<lb/>ſtra poſthac conſtabit) progredientes divergent; </s>
  <s xml:id="echoid-s705" xml:space="preserve">erítque proptereà <lb/>ang. </s>
  <s xml:id="echoid-s706" xml:space="preserve">QPC &amp;</s>
  <s xml:id="echoid-s707" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s708" xml:space="preserve">GFC; </s>
  <s xml:id="echoid-s709" xml:space="preserve">quare radius QP medium BC penetrabit, <lb/>ac refringetur, puta in PN; </s>
  <s xml:id="echoid-s710" xml:space="preserve">liquebit autem è dicendis refractos FM, <lb/>PN à ſe divergere; </s>
  <s xml:id="echoid-s711" xml:space="preserve">Hinc jam radiis MF, NP interpoſitum objectum <lb/>radiis OG, OQ interjectum apparebit, velut ad μ ν, ſitu neutiquam <lb/>immutatum. </s>
  <s xml:id="echoid-s712" xml:space="preserve">Rurſus autem ab oculi dicto centro prodeat alter radius <lb/>OK, cujus refractus ſit KI; </s>
  <s xml:id="echoid-s713" xml:space="preserve">hìc itaque rurſum à GF diverget, ac <lb/>indè erit ang. </s>
  <s xml:id="echoid-s714" xml:space="preserve">KIF &amp;</s>
  <s xml:id="echoid-s715" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s716" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s717" xml:space="preserve">GFC; </s>
  <s xml:id="echoid-s718" xml:space="preserve">adeóque KI minimè penetrabit <lb/>medium BC, at reflectetur, puta in IH; </s>
  <s xml:id="echoid-s719" xml:space="preserve">tum IH refringatur in HS. </s>
  <s xml:id="echoid-s720" xml:space="preserve"><lb/>Ergò jam radiis ER, HS interjacens objectum puta RS radiis OG, <lb/>OK interjectum cernetur, velut ad ρ σ, ſitu partium everſo. </s>
  <s xml:id="echoid-s721" xml:space="preserve">Conſe-<lb/>quuntur hæc doctrinam noſtram, &amp; </s>
  <s xml:id="echoid-s722" xml:space="preserve">experientiæ liquidò conſentanea <lb/>deprehendentur, quin &amp; </s>
  <s xml:id="echoid-s723" xml:space="preserve">obſervatu dignum erit, è duplici refractione <lb/>ſpectatum objectum MNIridis coloribus tinctum adparere (rubro <lb/>ſcilicet ad μ, cæruleo ad ν, croceo medium occupante) objecti verò <lb/>RS è duplici refractione, ſed reflectione tamen intercedente, appre-<lb/>henſi imaginem ρ σ colore nihil ab ipſo objecto differre. </s>
  <s xml:id="echoid-s724" xml:space="preserve">Quod ex eo <lb/>ſanè videtur evenire, quoniam ang FEB angulo FGC æquatur; </s>
  <s xml:id="echoid-s725" xml:space="preserve"><lb/>adeóque radius KO non aliter è vitro exit, quam RE ingreſſus eſt;</s>
  <s xml:id="echoid-s726" xml:space="preserve">
<pb o="19" file="0037" n="37" rhead=""/>
ſeu quicquid retractio ad E effecit, id refractio ad K retexit, radium in <lb/>ſtatum reſtituens, ei quem ab origine habuit non diſparem. </s>
  <s xml:id="echoid-s727" xml:space="preserve">Verùm <lb/>hæc non eſt hujus loci penitiùs excutere, Saltem obſervari meretur hoc <lb/>præcipuum, ut arbitror, in priſmate Phænomenon.</s>
  <s xml:id="echoid-s728" xml:space="preserve"/>
</p>
<div xml:id="echoid-div18" type="float" level="2" n="6">
<note position="left" xlink:label="note-0036-02" xlink:href="note-0036-02a" xml:space="preserve">Fig. 13.</note>
</div>
<p>
  <s xml:id="echoid-s729" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s730" xml:space="preserve">Hæc, inquam, cùm ab experientia confirmentur, neque tamen <lb/>ei magis, quam ratiociniis noſtris conſentiant, cauſis tamen adſcribun-<lb/>tur (à quibuſdam) nedum diverſis, at prorſus adverſis. </s>
  <s xml:id="echoid-s731" xml:space="preserve">Quòd enim <lb/>vitrum _e. </s>
  <s xml:id="echoid-s732" xml:space="preserve">g_. </s>
  <s xml:id="echoid-s733" xml:space="preserve">radios intra corpus ſuum receptos, ejúſque poſticæ ſuper-<lb/>ficiei obliquius incidentes retrocedere cogat, hujuſmodi rationem exhi-<lb/>bent: </s>
  <s xml:id="echoid-s734" xml:space="preserve">aiunt vitrum radios faciliùs admittere, vel tranſmittere quàm <lb/>aërem; </s>
  <s xml:id="echoid-s735" xml:space="preserve">quin addunt aërem vi reflectendi prævalidâ pollere; </s>
  <s xml:id="echoid-s736" xml:space="preserve">quódque <lb/>proinde qui poſt vitrum adventanti radio fit obvius aër cum reverbe-<lb/>rat. </s>
  <s xml:id="echoid-s737" xml:space="preserve">Quæ ratio mihi non adblanditur. </s>
  <s xml:id="echoid-s738" xml:space="preserve">Nam imprimis rei naturæ mi-<lb/>nùs conſentanea videtur. </s>
  <s xml:id="echoid-s739" xml:space="preserve">Quum enim aëris corpus ex æthere puro <lb/>maximam partem, è corpuſculis terrenis, &amp; </s>
  <s xml:id="echoid-s740" xml:space="preserve">ex halitibus aqueis con-<lb/>ſtare totum videatur; </s>
  <s xml:id="echoid-s741" xml:space="preserve">ex his partibus æther, opinor, non minùs <lb/>promptè quàm vitrum radios tranſmitti; </s>
  <s xml:id="echoid-s742" xml:space="preserve">aqua vero ſaltem (ex illo-<lb/>rum, quibuſcum diſputamus, ſententiâ) tantillo difficiliùs; </s>
  <s xml:id="echoid-s743" xml:space="preserve">adeóque <lb/>neutiquam harum alterutri dicta reflectio jure videtur attribuenda; <lb/></s>
  <s xml:id="echoid-s744" xml:space="preserve">terreſtres autem particulæ (quæ præ reliquis etiam pauciores videntur, <lb/>&amp; </s>
  <s xml:id="echoid-s745" xml:space="preserve">rariùs interſperſæ, præſertim in aëre ſudo ſummíſque; </s>
  <s xml:id="echoid-s746" xml:space="preserve">montium ex-<lb/>celſorum jugis) ſunt opacæ, nec lucem admittunt, aſt eam in omni-<lb/>cunque pariter incidentiâ rejiciunt; </s>
  <s xml:id="echoid-s747" xml:space="preserve">adeò ut nec ad has quam reſpici-<lb/>mus lucis inflectio propriè ſpectet; </s>
  <s xml:id="echoid-s748" xml:space="preserve">ergò nihil ſubeſſe videtur cauſæ, cur <lb/>aër (præ vitro) ingruenti luci potentiùs obſiſtat. </s>
  <s xml:id="echoid-s749" xml:space="preserve">Addo; </s>
  <s xml:id="echoid-s750" xml:space="preserve">ſi talis aëri vis <lb/>reflexiva competat, &amp; </s>
  <s xml:id="echoid-s751" xml:space="preserve">vera ſit, quam hi Philoſophi memorato Phæ-<lb/>nomeno cauſam adſignant, conſequi videtur, ut nulla prope Horizon-<lb/>tem Stella conſpici poſſit (admiſſo ſaltem hoc, non inverecundo reor, <lb/>ut pleriſque viſum erit, poftulato; </s>
  <s xml:id="echoid-s752" xml:space="preserve">quòd purus Æther, naturale lucis <lb/>vehiculum, aëreæ regioni ſuprajectus haud mînùs facile quàm vitrum <lb/>aut aqua radios intromittet.) </s>
  <s xml:id="echoid-s753" xml:space="preserve">Sit enim C terræ centrum, O oculus, <lb/>S viſibile punctum longinquum, ceu ſtella, ſitum in Horizonte; </s>
  <s xml:id="echoid-s754" xml:space="preserve">per <lb/>
<anchor type="note" xlink:label="note-0037-01a" xlink:href="note-0037-01"/>
puncta verò C, O, S trajectum concipiatur planum faciens in terræ ſu-<lb/>perficie circùlum OP; </s>
  <s xml:id="echoid-s755" xml:space="preserve">in atmoſphæræ, vel aëris circumfuſi, extima <lb/>ſuperficie circumferentiam ZNM; </s>
  <s xml:id="echoid-s756" xml:space="preserve">itaque cum radius quilibet, ut <lb/>SM, vel SN (Horizontalis puta, vel eò ſuperior) in ſuperficiem MZ <lb/>cadens ei incidat perquam obliquè (niſi ſaltem atmoſphæræ Semidi-<lb/>ameter CZ præ telluris Semidiametro CO dicatur enormiter, &amp; </s>
  <s xml:id="echoid-s757" xml:space="preserve">in-<lb/>credibiliter magna) cùm &amp; </s>
  <s xml:id="echoid-s758" xml:space="preserve">aer idcircò, juxta ſententiam quam ex-
<pb o="20" file="0038" n="38" rhead=""/>
pendimus, lucem admittere non debeat; </s>
  <s xml:id="echoid-s759" xml:space="preserve">omnino dicti radii SM, SN, <lb/>&amp; </s>
  <s xml:id="echoid-s760" xml:space="preserve">conſimiles reflectentur, &amp; </s>
  <s xml:id="echoid-s761" xml:space="preserve">extra atmoſphæram procul abeuntes <lb/>oculum non pertingent. </s>
  <s xml:id="echoid-s762" xml:space="preserve">Quinimò ſatìs conſtare videtur exhinc, quòd <lb/>radii quales SN ut viſum afficere queant, aut accedere, verſus per-<lb/>pendicularem NC refringi debent; </s>
  <s xml:id="echoid-s763" xml:space="preserve">id quod adverſariæ Hypotheſi <lb/>pariter adverſatur. </s>
  <s xml:id="echoid-s764" xml:space="preserve">Verùm hæc obiter, ac in trancurſu dicta ſunto.</s>
  <s xml:id="echoid-s765" xml:space="preserve"/>
</p>
<div xml:id="echoid-div19" type="float" level="2" n="7">
<note position="right" xlink:label="note-0037-01" xlink:href="note-0037-01a" xml:space="preserve">Fig. 14.</note>
</div>
<p>
  <s xml:id="echoid-s766" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s767" xml:space="preserve">Porrò, ſubnotandum eſt, quoad binos caſus ſuprâ tractatos, <lb/>cùm duo media, diverſimoda comparandò duæ ſe repreſentent propor-<lb/>tiones, altera, terminorum ſitum tranſponendo, alterius inverſa, re-<lb/>fractionum ideò menſuras (quoad hæc) iiſdem terminis deſignabiles <lb/>ordine permutari. </s>
  <s xml:id="echoid-s768" xml:space="preserve">Ut ſi in primo caſu ſinus rectus anguli incidentis ſe <lb/>habeat ad ſinum rectum anguli refracti, ſicut A ad B, in ſecundo ſinus <lb/>incidentis ad ſinum refracti ſe inverſè habebit ut B ad A; </s>
  <s xml:id="echoid-s769" xml:space="preserve">nimirum <lb/>in præcedentibus ſiguris, quanto ZD major eſt quàm ZB, in prima <lb/>Hypotheſi; </s>
  <s xml:id="echoid-s770" xml:space="preserve">tanto conſtat ZD minorem eſſe quàm ZB, in ſecunda; <lb/></s>
  <s xml:id="echoid-s771" xml:space="preserve">mediis ſcilicet iiſdem permanentibus.</s>
  <s xml:id="echoid-s772" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s773" xml:space="preserve">X. </s>
  <s xml:id="echoid-s774" xml:space="preserve">Denique, cùm medii cui radius impingit ſuperficiem hactenus <lb/>adſumpſerimus planam, advertendum ſupereſt, quamvìs illa curva ſit, <lb/>eodem tamen abſque ſenſibili diſcrimine ſeſe modo rem habere, ac ſi <lb/>plano curvam ſuperficiem iſthic, ubi radius occurrit, contingenti im-<lb/>pingeret. </s>
  <s xml:id="echoid-s775" xml:space="preserve">Incidat nempe radius ABCD in curvam lineam QBR, <lb/>quam ad incidentiæ punctum B tangat recta EF. </s>
  <s xml:id="echoid-s776" xml:space="preserve">Prorſuseodem mo-<lb/>
<anchor type="note" xlink:label="note-0038-01a" xlink:href="note-0038-01"/>
do refringetur iſte radius ad curvam QBR, quo ad rectam EF, niſi <lb/>quòd iſthic arcus D δ in rariori medio decurſus tangentem aliquouſque <lb/>prætergreditur. </s>
  <s xml:id="echoid-s777" xml:space="preserve">Id quod eximiam radii ſubtilitatem conſiderando, <lb/>quámque perexiguo diſtet intervallo punctum δ à curvæ vertice B, <lb/>nullam omnino ſenſibilem (imò nec imaginabilem) inducet differen-<lb/>tiam. </s>
  <s xml:id="echoid-s778" xml:space="preserve">Quantillus enim iſte circulus eſſe debet, in quo chorda B δ, ra-<lb/>dii latitudine paullo major, arcum ſubtendet aliqua cum ejus ſenſibili <lb/>parte comparabilem? </s>
  <s xml:id="echoid-s779" xml:space="preserve">poteſt igitur angulus ZB δ æqualis ſupponi <lb/>angulo ZBF; </s>
  <s xml:id="echoid-s780" xml:space="preserve">quo conceſſo reliqua fluent eodem tenore, quo præ-<lb/>cedentia. </s>
  <s xml:id="echoid-s781" xml:space="preserve">Quin unà rationem exhibuimus ſuppoſitionis, quæ paſſim <lb/>ab Opticis accipitur; </s>
  <s xml:id="echoid-s782" xml:space="preserve">ità tamen precariò, non ut ſubinde nullum <lb/>in audientibus ſcrupulum relinquat; </s>
  <s xml:id="echoid-s783" xml:space="preserve">nec ut ſemper adſenſu firmo <lb/>concedatur.</s>
  <s xml:id="echoid-s784" xml:space="preserve"/>
</p>
<div xml:id="echoid-div20" type="float" level="2" n="8">
<note position="right" xlink:label="note-0038-01" xlink:href="note-0038-01a" xml:space="preserve">Fig. 15.</note>
</div>
<p>
  <s xml:id="echoid-s785" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s786" xml:space="preserve">Ità primarias iſtas circa radiorum inflectionem Hypotheſes-<lb/>(vel Axiomata malitis, aut Theoremata) quibus omnis incumbit Optica
<pb o="21" file="0039" n="39" rhead=""/>
cujuſcunque generis ſcientia, qualitercunque declarare ſtuduimus, &amp; </s>
  <s xml:id="echoid-s787" xml:space="preserve"><lb/>è principiis admodum affinibus elicere; </s>
  <s xml:id="echoid-s788" xml:space="preserve">modo, meâ ſententiâ ſaltem, <lb/>omnium qui legenti ſe vel cogitanti ſuggeſſerunt, ſimillimo veri, cúm-<lb/>que tam rationibus Mechanicis, quàm experimentis Phyſicis, &amp; </s>
  <s xml:id="echoid-s789" xml:space="preserve">cùm <lb/>ipſa rerum natura congruentiſſimo. </s>
  <s xml:id="echoid-s790" xml:space="preserve">Neque nulla mihi tunc oborieba-<lb/>tur voluptas, cùm poſtquam inter alios iſta lucis ſymptomata expli-<lb/>candi modos hic ipſe ſemet ingeſſerat, eum examini ſubjiciens, Geo-<lb/>metriæ legibus (aliquanto ſanè præter expectationem) adeo quadran-<lb/>tem comperiſſem. </s>
  <s xml:id="echoid-s791" xml:space="preserve">Meæ tamen eum tam fuſe4; </s>
  <s xml:id="echoid-s792" xml:space="preserve">diducendi peperciffem <lb/>operæ, ſi quæ doctiſſimus _Maignanus_ hiſce conformia, luculentius <lb/>quidem opinor &amp; </s>
  <s xml:id="echoid-s793" xml:space="preserve">accuratius, pertractavit, priuſquam hæc aggrederer <lb/>contigiſſet inſpexiſſe; </s>
  <s xml:id="echoid-s794" xml:space="preserve">penes quem extare multa nil dubitem (nec enim <lb/>eum adhuc curioſiùs evolvi) ſupplendis his, &amp; </s>
  <s xml:id="echoid-s795" xml:space="preserve">confirmandis accommo-<lb/>data. </s>
  <s xml:id="echoid-s796" xml:space="preserve">Porro fuit etiam animus, alias, quæ plurimæ traduntur, horum <lb/>rationes percenſere, ac perſtringere (quarum mihi nonnullæ craſſâ <lb/>petitione laborare; </s>
  <s xml:id="echoid-s797" xml:space="preserve">multùm aliæ à re propoſita abludere; </s>
  <s xml:id="echoid-s798" xml:space="preserve">quædam <lb/>animum ſubtilitate potius confundere, quàm vi conſtringere videban-<lb/>tur) verùm etiam huic exponendæ nimiſquam immoratus, hactenus in-<lb/>ſinuatis contentus, omnes tranſiliam; </s>
  <s xml:id="echoid-s799" xml:space="preserve">illius ſaltem eruditiſſimi viri <lb/>nefas fuerit non aſtipulari penitus, &amp; </s>
  <s xml:id="echoid-s800" xml:space="preserve">acquieſcere decreto; </s>
  <s xml:id="echoid-s801" xml:space="preserve">“qui, De-<lb/>"um unicum &amp; </s>
  <s xml:id="echoid-s802" xml:space="preserve">Optimum Naturæ Architectum, hanc (ait) legem ra-<lb/>"diis diverſa media permeantibus præſcripſiſſe; </s>
  <s xml:id="echoid-s803" xml:space="preserve">ut omnes omnino <lb/>"radii veri, &amp; </s>
  <s xml:id="echoid-s804" xml:space="preserve">apparentes eandem ſemper inter ſe ſervent analogiam. <lb/></s>
  <s xml:id="echoid-s805" xml:space="preserve">His, inquam, dimiſſis, ſuccedit ut è præſtructis emanantia quædam <lb/>Poriſmata ſubnectamus.</s>
  <s xml:id="echoid-s806" xml:space="preserve"/>
</p>
<pb o="22" file="0040" n="40"/>
</div>
<div xml:id="echoid-div22" type="section" level="1" n="11">
<head xml:id="echoid-head14" xml:space="preserve"><emph style="sc">Lect</emph>. III.</head>
<p>
  <s xml:id="echoid-s807" xml:space="preserve">I. </s>
  <s xml:id="echoid-s808" xml:space="preserve">HYpotheſes Opticæ primarias, &amp; </s>
  <s xml:id="echoid-s809" xml:space="preserve">fundamentales quaſi leges <lb/>expoſuimus hactenus, &amp; </s>
  <s xml:id="echoid-s810" xml:space="preserve">excuſſimus quomodocunque: </s>
  <s xml:id="echoid-s811" xml:space="preserve">Se-<lb/>quitur jam ut ex iis emergentia quædam (ad apparentiarum cauſas tam <lb/>verè quàm expeditè diſcernendas conducentia) ſubjungamus corollaria; <lb/></s>
  <s xml:id="echoid-s812" xml:space="preserve">de cæteris quæ faciliora videntur, aut uſum præ ſe ferunt potiſſimum <lb/>ſeligentes. </s>
  <s xml:id="echoid-s813" xml:space="preserve">Radios autem jam conſideramus, ut unicâ dimenſione præ-<lb/>ditos (ſiquidem reliquæ, quibus Phyſicè gaudent, parùm faciunt ad <lb/>computationes hîc inſtitutas) ut lineas, inquam ceu vulgò fit, rectas <lb/>concipimus a lucido quolibet aut aſpectabili puncto dimanantes. </s>
  <s xml:id="echoid-s814" xml:space="preserve">Quin <lb/>&amp; </s>
  <s xml:id="echoid-s815" xml:space="preserve">cum, hoc admiſſo, ſinguli cujuſque radii inflectio in ſuperficie per-<lb/>agatur ad planum inflectens recta (uti conſtat è præmiſſis) cùm &amp; </s>
  <s xml:id="echoid-s816" xml:space="preserve">no-<lb/>bis præſertim mox inſtitutum ſit ſingulorum punctorum radiationes <lb/>conſequentia ſymptomata ſic expendere, ut locos ipſorum apparen-<lb/>tes determinemus, oculi reſpectu centrum habentis in ejuſmodi <lb/>plano uſpiam conſtitutum; </s>
  <s xml:id="echoid-s817" xml:space="preserve">pro planis ubique rectas lineas, pro Sphæ-<lb/>ricis ſuperficiebus peripherias circulares, pro reliquis lineas reſpectivè <lb/>congruas, brevitati conſulentes &amp; </s>
  <s xml:id="echoid-s818" xml:space="preserve">perſpicuitati, ſubſtituemus. </s>
  <s xml:id="echoid-s819" xml:space="preserve">Porrò <lb/>cùm quo præciſè modo peragatur viſio, quibúſque prædita ſit affecti-<lb/>onibus adhuc expoſitum non ſit; </s>
  <s xml:id="echoid-s820" xml:space="preserve">Et de illa tamen ſubindè craſſius ali-<lb/>quid ac generalius dicendis intexere fortaſſis ex uſu fuerit, illa ſaltem <lb/>pervnlgata, poſt hac curioſiùs expendenda, jam προλεπτικῶς adſume-<lb/>mus; </s>
  <s xml:id="echoid-s821" xml:space="preserve">nempe: </s>
  <s xml:id="echoid-s822" xml:space="preserve">Viſibile punctum in illo radio ſitum apparere, qui pro-<lb/>cedens ab ipſo (directè vel inflexè) centrum oculi permeat; </s>
  <s xml:id="echoid-s823" xml:space="preserve">proindéq; </s>
  <s xml:id="echoid-s824" xml:space="preserve"><lb/>ſitum objectorum è radiorum ità tranſeuntium poſitione judicari. </s>
  <s xml:id="echoid-s825" xml:space="preserve">Ma-<lb/>jora, minora, vel æqualia videri objecta, prout ipſorum extrema pun-<lb/>cta radiis cernuntur angulos ad oculi centrum reſpectivè majores, mi-<lb/>nores, aut æquales conſtituentibus; </s>
  <s xml:id="echoid-s826" xml:space="preserve">diſtinctam unius cujuſque puncti <lb/>viſionem radiis effici modo naturali, hoc eſt, divergentèr, oculo illa-<lb/>bentibus; </s>
  <s xml:id="echoid-s827" xml:space="preserve">Et ſiqua ſunt his agnata pariter obvia, ſeu manifeſta. </s>
  <s xml:id="echoid-s828" xml:space="preserve"><lb/>Quinetiam, verborum parci, vocabulis paſſim receptis &amp; </s>
  <s xml:id="echoid-s829" xml:space="preserve">uſitatis defi-<lb/>niendis aut explicandis abſtinentes, ipſorum ſupponimus intelle-
<pb o="23" file="0041" n="41" rhead=""/>
ctum. </s>
  <s xml:id="echoid-s830" xml:space="preserve">His utcunque majoris evidentiæ cauſâ, prælibatis, ad corolla-<lb/>ria quæ diximus expromenda nos conferemus è veſtigio.</s>
  <s xml:id="echoid-s831" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s832" xml:space="preserve">II. </s>
  <s xml:id="echoid-s833" xml:space="preserve">Imprimis autem (poſthac quidem in decurſu, quoad plures ſibi <lb/>parallelos, aut ab eodem puncto divergentes (velin idem convergen-<lb/>tes) &amp; </s>
  <s xml:id="echoid-s834" xml:space="preserve">huic vel illi ſingulari, quæ tractanda veniet, ſuperficiei inciden-<lb/>tes (radios, ſingularia quotvis inflexos deſignandi compendia, radia-<lb/>tionibus organicè examinandis profutura, tradituri) generales nunc <lb/>aliquos incidenti cuivis propoſito competentem inflexum aſſignandi <lb/>modos proponemus; </s>
  <s xml:id="echoid-s835" xml:space="preserve">quorum adhiberi poſſit, qui rei natæ videbitur <lb/>accommodatior. </s>
  <s xml:id="echoid-s836" xml:space="preserve">Pro reflectione. </s>
  <s xml:id="echoid-s837" xml:space="preserve">Incidat radius AB ad B; </s>
  <s xml:id="echoid-s838" xml:space="preserve">Et per <lb/>B ducatur QB reflectenti perpendicularis; </s>
  <s xml:id="echoid-s839" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s840" xml:space="preserve">fiat ang. </s>
  <s xml:id="echoid-s841" xml:space="preserve">_a_ BQ = ang. <lb/></s>
  <s xml:id="echoid-s842" xml:space="preserve">ABQ, vel per B ducta ſit EF reflectentem tangens; </s>
  <s xml:id="echoid-s843" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s844" xml:space="preserve">fiat ang. </s>
  <s xml:id="echoid-s845" xml:space="preserve"><lb/>a BF = ang. </s>
  <s xml:id="echoid-s846" xml:space="preserve">ABE; </s>
  <s xml:id="echoid-s847" xml:space="preserve">liquetque factum eſſe, modo utrovis, quod <lb/>
<anchor type="note" xlink:label="note-0041-01a" xlink:href="note-0041-01"/>
requirebatur, Pro refractione vero; </s>
  <s xml:id="echoid-s848" xml:space="preserve">ducatur QB refringenti per-<lb/>pendicularis, &amp; </s>
  <s xml:id="echoid-s849" xml:space="preserve">ſuper diametrum (in hac liberè ſumptam) QB de-<lb/>ſcribatur ſemicirculus, incidentem AB ſecans in R; </s>
  <s xml:id="echoid-s850" xml:space="preserve">tum adjunctâ <lb/>QR, factóque I. </s>
  <s xml:id="echoid-s851" xml:space="preserve">R :</s>
  <s xml:id="echoid-s852" xml:space="preserve">: QR. </s>
  <s xml:id="echoid-s853" xml:space="preserve">T (terminis autem I, R hìc &amp; </s>
  <s xml:id="echoid-s854" xml:space="preserve">dehinc <lb/>perpetuò proportio refractiones metiens indignitatur) circulo QRB <lb/>adaptetur QS ipſam T exæquans; </s>
  <s xml:id="echoid-s855" xml:space="preserve">erit connexa SB protracta nempe) <lb/>incidentis AB refracta. </s>
  <s xml:id="echoid-s856" xml:space="preserve">? </s>
  <s xml:id="echoid-s857" xml:space="preserve">Vel: </s>
  <s xml:id="echoid-s858" xml:space="preserve">perincidentiæ punctum B ducatur EF <lb/>refringentem contingens; </s>
  <s xml:id="echoid-s859" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s860" xml:space="preserve">in hac utcunque ſumpta BK ſit circuli <lb/>diameter, incidentem AB ſecantis ad R; </s>
  <s xml:id="echoid-s861" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s862" xml:space="preserve">fiat I. </s>
  <s xml:id="echoid-s863" xml:space="preserve">R:</s>
  <s xml:id="echoid-s864" xml:space="preserve">: BR. </s>
  <s xml:id="echoid-s865" xml:space="preserve">T; <lb/></s>
  <s xml:id="echoid-s866" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s867" xml:space="preserve">adaptetur BS = T; </s>
  <s xml:id="echoid-s868" xml:space="preserve">erit SB _a_ ipſius AB refractus: </s>
  <s xml:id="echoid-s869" xml:space="preserve">Vel demum: </s>
  <s xml:id="echoid-s870" xml:space="preserve"><lb/>In ipſa AB ſumptâ utcunque diametro RB, ſuper hac deſcriptus cir-<lb/>culus ſecet perpendicularem QB ad Q; </s>
  <s xml:id="echoid-s871" xml:space="preserve">vel tangentem EF in K; </s>
  <s xml:id="echoid-s872" xml:space="preserve"><lb/>fiátque I. </s>
  <s xml:id="echoid-s873" xml:space="preserve">R :</s>
  <s xml:id="echoid-s874" xml:space="preserve">: BK. </s>
  <s xml:id="echoid-s875" xml:space="preserve">T. </s>
  <s xml:id="echoid-s876" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s877" xml:space="preserve">adaptetur QS = T; </s>
  <s xml:id="echoid-s878" xml:space="preserve">erit rurſus SB _a_ inci-<lb/>dentis AB refractus. </s>
  <s xml:id="echoid-s879" xml:space="preserve">quorum ratio è poſitis inflectionum legibus ad-<lb/>modum eſt manifeſta. </s>
  <s xml:id="echoid-s880" xml:space="preserve">verba piget impendere.</s>
  <s xml:id="echoid-s881" xml:space="preserve"/>
</p>
<div xml:id="echoid-div22" type="float" level="2" n="1">
<note position="right" xlink:label="note-0041-01" xlink:href="note-0041-01a" xml:space="preserve">Fig. 16. <lb/>Fig. 17, 18, <lb/>19.</note>
</div>
<p>
  <s xml:id="echoid-s882" xml:space="preserve">III. </s>
  <s xml:id="echoid-s883" xml:space="preserve">Radii cujuſvis incidentis inflexus inflexi viciſſim incidens <lb/>evadet.</s>
  <s xml:id="echoid-s884" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s885" xml:space="preserve">Hoc plerique, diverſè paullò prolatum, accipiunt, aut poſtulant. <lb/></s>
  <s xml:id="echoid-s886" xml:space="preserve">
<anchor type="note" xlink:label="note-0041-02a" xlink:href="note-0041-02"/>
è præmiſtis autem facillimè colligitur. </s>
  <s xml:id="echoid-s887" xml:space="preserve">Idque potius methodi gratiâ <lb/>(ſicut &amp; </s>
  <s xml:id="echoid-s888" xml:space="preserve">nonnulla quæ ſequentur) quàm quia res meretur, oſtende-<lb/>mus. </s>
  <s xml:id="echoid-s889" xml:space="preserve">Proreflectione; </s>
  <s xml:id="echoid-s890" xml:space="preserve">Radius AB ſpeculo EF impingens reflectatur <lb/>
<anchor type="note" xlink:label="note-0041-03a" xlink:href="note-0041-03"/>
in B _a_; </s>
  <s xml:id="echoid-s891" xml:space="preserve">dico radium B _a_ permutatim in BA reflecti. </s>
  <s xml:id="echoid-s892" xml:space="preserve">Nam quoniam <lb/>AB incidens reflectitur in B _a_, erit ang. </s>
  <s xml:id="echoid-s893" xml:space="preserve">_a_ BF æ qualis angulo ABE. <lb/></s>
  <s xml:id="echoid-s894" xml:space="preserve">Poſito jam _a_ B incidere, etiam angulus quem facit ejus reflexus cum <lb/>BE æquabitur angulo _a_ BF; </s>
  <s xml:id="echoid-s895" xml:space="preserve">proinde non alius erit ab ipſo ABE <lb/>quare BA ipſius _a_ B reflexus erit.</s>
  <s xml:id="echoid-s896" xml:space="preserve"/>
</p>
<div xml:id="echoid-div23" type="float" level="2" n="2">
<note position="right" xlink:label="note-0041-02" xlink:href="note-0041-02a" xml:space="preserve">_Albaz_. VII. 4. <lb/>_Herig. Catop_. <lb/>_Axiom._ 2.</note>
<note position="right" xlink:label="note-0041-03" xlink:href="note-0041-03a" xml:space="preserve">Fig. 1, 16.</note>
</div>
<pb o="24" file="0042" n="42" rhead=""/>
<p>
  <s xml:id="echoid-s897" xml:space="preserve">Pro refractione vero: </s>
  <s xml:id="echoid-s898" xml:space="preserve">Incidat radius AB medio EF in B; </s>
  <s xml:id="echoid-s899" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s900" xml:space="preserve">iſthic <lb/>refringatur in B _a_ dico permutatim radium _a_ B regredientem in BA <lb/>refringi. </s>
  <s xml:id="echoid-s901" xml:space="preserve">Nam per occurſum B ducatur QBP media dirimenti EF <lb/>
<anchor type="note" xlink:label="note-0042-01a" xlink:href="note-0042-01"/>
perpendicularis; </s>
  <s xml:id="echoid-s902" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s903" xml:space="preserve">in hac utcunque ſumpto puncto P ducatur PG <lb/>ad AB protractam perpendicularis, ut &amp; </s>
  <s xml:id="echoid-s904" xml:space="preserve">PH ad B _a_; </s>
  <s xml:id="echoid-s905" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s906" xml:space="preserve">produca-<lb/>tur _a_ BS. </s>
  <s xml:id="echoid-s907" xml:space="preserve">Eſt ergo PG ſinus rectus anguli incidentiæ ABQ ad <lb/>radium BP; </s>
  <s xml:id="echoid-s908" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s909" xml:space="preserve">PH ſinus anguli refracti QBS ad eundem radium <lb/>BP. </s>
  <s xml:id="echoid-s910" xml:space="preserve">Cùm ìtaque ratio PG ad PH refractionem metiatur è ſuperiori <lb/>medio factam in inferius; </s>
  <s xml:id="echoid-s911" xml:space="preserve">etiam viciſſim rectarum PH, PG propor-<lb/>tio refractionem determinabit ab inferiori medio factam in ſuperius. <lb/></s>
  <s xml:id="echoid-s912" xml:space="preserve">Unde ſi radius _a_ B jam ponatur incidens; </s>
  <s xml:id="echoid-s913" xml:space="preserve">cùm ſint PH, PG recti ſi-<lb/>nus anguli incidentiæ PBH, &amp; </s>
  <s xml:id="echoid-s914" xml:space="preserve">anguli PBG, liquidum eſt ipſam HB <lb/>in BA refringi.</s>
  <s xml:id="echoid-s915" xml:space="preserve"/>
</p>
<div xml:id="echoid-div24" type="float" level="2" n="3">
<note position="left" xlink:label="note-0042-01" xlink:href="note-0042-01a" xml:space="preserve">Fig. 20.</note>
</div>
<p>
  <s xml:id="echoid-s916" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s917" xml:space="preserve">Angulo incidentiæ majori major competit angulus inflexus. <lb/></s>
  <s xml:id="echoid-s918" xml:space="preserve">(Angulum inflexum vocito, qui à perpendiculari continetur &amp; </s>
  <s xml:id="echoid-s919" xml:space="preserve">in-<lb/>flexo; </s>
  <s xml:id="echoid-s920" xml:space="preserve">is proinde reſpectivè dicitur angulus reflexus, vel refractus. </s>
  <s xml:id="echoid-s921" xml:space="preserve"><lb/>Angulus autem inflectionis (hoc eſt reflectionis reſpectivè, vel refracti-<lb/>onis) appellatur is, qui comprehenditur ab incidente &amp; </s>
  <s xml:id="echoid-s922" xml:space="preserve">inflexo; </s>
  <s xml:id="echoid-s923" xml:space="preserve">inci-<lb/>dentiæ verò, nè quis ſecùs accipiat, apud nos angulus eſt, quem con-<lb/>tinent incidens &amp; </s>
  <s xml:id="echoid-s924" xml:space="preserve">perpendicularis.) </s>
  <s xml:id="echoid-s925" xml:space="preserve">Quod propoſitum ſpectat effatum, <lb/>id è poſitis principiis manifeſtè conſectatur. </s>
  <s xml:id="echoid-s926" xml:space="preserve">Etenim in reflectione <lb/>ipſi anguli reflexi angulis incidentiæ proportionales ſunt; </s>
  <s xml:id="echoid-s927" xml:space="preserve">in refracti-<lb/>one ſaltem recti ſinus angulorum refractorum ſinubus angulorum in-<lb/>cidentiæ proportionantur. </s>
  <s xml:id="echoid-s928" xml:space="preserve">Unde liquido conſtat propoſitum: </s>
  <s xml:id="echoid-s929" xml:space="preserve">quorſum <lb/>verba, quorſum Schemata multiplicem?</s>
  <s xml:id="echoid-s930" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s931" xml:space="preserve">V. </s>
  <s xml:id="echoid-s932" xml:space="preserve">Cùm incidentes ad ſuperficiem mediam ſeſe decuſſant, iidem <lb/>ſeſe inflectionem paſſi decuſſabunt, eodem ordine ſervato, quem directè <lb/>progedientes habuiſſent (utique ſic ut perpendiculari poſt inflectionem <lb/>propior incedat qui propior antea fuit.)</s>
  <s xml:id="echoid-s933" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s934" xml:space="preserve">Hæc propoſitio reverà non differt à præcedente; </s>
  <s xml:id="echoid-s935" xml:space="preserve">quò demirer <lb/>
<anchor type="note" xlink:label="note-0042-02a" xlink:href="note-0042-02"/>
eam à non nemine principia noſtra uſurpante aliunde compro-<lb/>bari.</s>
  <s xml:id="echoid-s936" xml:space="preserve"/>
</p>
<div xml:id="echoid-div25" type="float" level="2" n="4">
<note position="left" xlink:label="note-0042-02" xlink:href="note-0042-02a" xml:space="preserve">_Herig. Diop. 8_.</note>
</div>
<p>
  <s xml:id="echoid-s937" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s938" xml:space="preserve">Angulo incidentiæ majori major convenit angul@s inflectionis. <lb/></s>
  <s xml:id="echoid-s939" xml:space="preserve">Quoad refiectionem, res extra dubium evidens eſt; </s>
  <s xml:id="echoid-s940" xml:space="preserve">angulus enim refle-<lb/>ctionis incidentiæ majori conveniens eum planè continet, qui minori <lb/>incidentiæ reſpondet. </s>
  <s xml:id="echoid-s941" xml:space="preserve">Pro refractione vero: </s>
  <s xml:id="echoid-s942" xml:space="preserve">ſit recta QBP refrin-<lb/>genti perpendicularis; </s>
  <s xml:id="echoid-s943" xml:space="preserve">incidant autem radii ABG, DBH (ſcilicet AB
<pb o="25" file="0043" n="43" rhead=""/>
obliquiùs quàm DB.) </s>
  <s xml:id="echoid-s944" xml:space="preserve">Horum verò refracti ſint B _a_, Bδ; </s>
  <s xml:id="echoid-s945" xml:space="preserve">dico an-<lb/>
<anchor type="note" xlink:label="note-0043-01a" xlink:href="note-0043-01"/>
gulum β B _a_ majorem eſſe angulo HB δ. </s>
  <s xml:id="echoid-s946" xml:space="preserve">Nam ad BP in perpendicu-<lb/>lari liberè ſumptam diametrum conſtituatur ſemicirculus BGP; </s>
  <s xml:id="echoid-s947" xml:space="preserve">cui <lb/>occurrant ipſæ AB, DB protractæ ad G, H; </s>
  <s xml:id="echoid-s948" xml:space="preserve">nec non ipſæ B _a_, B δ <lb/>punctis _a_, δ. </s>
  <s xml:id="echoid-s949" xml:space="preserve">Fiat autem angulus GBK æqualis angulo HBδ, vel <lb/>arcus GK arcui Hδ; </s>
  <s xml:id="echoid-s950" xml:space="preserve">connectatur etiam rècta δ G, ſecans ipſam PK <lb/>in X; </s>
  <s xml:id="echoid-s951" xml:space="preserve">ducatnurque denuò ſubtenſæ G δ, H δ. </s>
  <s xml:id="echoid-s952" xml:space="preserve">Jam ob angulos PG δ, <lb/>PH δ pares (arcui quippe P δ inſiſtentes ambos) &amp; </s>
  <s xml:id="echoid-s953" xml:space="preserve">angulos GPK, <lb/>
<anchor type="note" xlink:label="note-0043-02a" xlink:href="note-0043-02"/>
HP δ ex conſtructione quoque pares, erunt triangula GPX, <lb/>HP δ inter ſe ſimilia. </s>
  <s xml:id="echoid-s954" xml:space="preserve">Quapropter erit PG. </s>
  <s xml:id="echoid-s955" xml:space="preserve">PX :</s>
  <s xml:id="echoid-s956" xml:space="preserve">: PH. </s>
  <s xml:id="echoid-s957" xml:space="preserve">P δ. </s>
  <s xml:id="echoid-s958" xml:space="preserve">eſt <lb/>autem, è lege refractionum PH. </s>
  <s xml:id="echoid-s959" xml:space="preserve">P δ :</s>
  <s xml:id="echoid-s960" xml:space="preserve">: PG. </s>
  <s xml:id="echoid-s961" xml:space="preserve">P _a_. </s>
  <s xml:id="echoid-s962" xml:space="preserve">quare PG. </s>
  <s xml:id="echoid-s963" xml:space="preserve">PX :</s>
  <s xml:id="echoid-s964" xml:space="preserve">: <lb/>PG. </s>
  <s xml:id="echoid-s965" xml:space="preserve">P _a_: </s>
  <s xml:id="echoid-s966" xml:space="preserve">unde PX = P _a_. </s>
  <s xml:id="echoid-s967" xml:space="preserve">eſt autem PX minor quàm PK (quia <lb/>tota ſubtenſa G δ intra circulum jacet.) </s>
  <s xml:id="echoid-s968" xml:space="preserve">Quare P _a_ minor eſt quàm <lb/>PK; </s>
  <s xml:id="echoid-s969" xml:space="preserve">adeóque PK ſecabit angulum GP _a_. </s>
  <s xml:id="echoid-s970" xml:space="preserve">quamobrem arcùs G _a_ ma-<lb/>jor erit arcu GK, hoc eſt arcu H δ. </s>
  <s xml:id="echoid-s971" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s972" xml:space="preserve">idcircò major erit angulus <lb/>GB _a_ angulo HB δ: </s>
  <s xml:id="echoid-s973" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s974" xml:space="preserve">E. </s>
  <s xml:id="echoid-s975" xml:space="preserve">D.</s>
  <s xml:id="echoid-s976" xml:space="preserve"/>
</p>
<div xml:id="echoid-div26" type="float" level="2" n="5">
<note position="right" xlink:label="note-0043-01" xlink:href="note-0043-01a" xml:space="preserve">Fig. 21.</note>
<note position="right" xlink:label="note-0043-02" xlink:href="note-0043-02a" xml:space="preserve">Fig. 22.</note>
</div>
<p>
  <s xml:id="echoid-s977" xml:space="preserve">Procedit hæc demonſtratio quoad caſum, ubi I &amp;</s>
  <s xml:id="echoid-s978" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s979" xml:space="preserve">R (vel cùm ra-<lb/>dius è medio rariori denſius ingreditur) at exinde quoad alterum quo-<lb/>que caſum facilè deducitur concluſio. </s>
  <s xml:id="echoid-s980" xml:space="preserve">Nam ſi viciſſim _a_ B, δ B con-<lb/>cipiantur incidentes, erunt ipſæ BA, BD earum refractæ; </s>
  <s xml:id="echoid-s981" xml:space="preserve">ac etiam-<lb/>num anguli _a_ BG, δ BH erunt anguli refracti.</s>
  <s xml:id="echoid-s982" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s983" xml:space="preserve">Hujuſce Theorematis apud _Herigonium_ habetur alia demonſtra-<lb/>
<anchor type="note" xlink:label="note-0043-03a" xlink:href="note-0043-03"/>
tio. </s>
  <s xml:id="echoid-s984" xml:space="preserve">Confer ſodes, &amp; </s>
  <s xml:id="echoid-s985" xml:space="preserve">utramvis elige. </s>
  <s xml:id="echoid-s986" xml:space="preserve">No3 quam res obtulit <lb/>poſuimus.</s>
  <s xml:id="echoid-s987" xml:space="preserve"/>
</p>
<div xml:id="echoid-div27" type="float" level="2" n="6">
<note position="right" xlink:label="note-0043-03" xlink:href="note-0043-03a" xml:space="preserve">_Diopt<unsure/>. Prop@.4_.</note>
</div>
<p>
  <s xml:id="echoid-s988" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s989" xml:space="preserve">In iſto refractionis caſu, quum I minor eſt quàm R, ſi anguli <lb/>
<anchor type="note" xlink:label="note-0043-04a" xlink:href="note-0043-04"/>
incidentiæ, puta anguli DBQ, rectus ſinus PH, ad ſinum totum ſe <lb/>habeat ut I ad R; </s>
  <s xml:id="echoid-s990" xml:space="preserve">nullus incidente DB obliquior radius medium EF <lb/>refractus ingredietur, aut penetrabit.</s>
  <s xml:id="echoid-s991" xml:space="preserve"/>
</p>
<div xml:id="echoid-div28" type="float" level="2" n="7">
<note position="right" xlink:label="note-0043-04" xlink:href="note-0043-04a" xml:space="preserve">Fig. 23.</note>
</div>
<p>
  <s xml:id="echoid-s992" xml:space="preserve">Nam penerret (ſi fieri poteſt) obliquioris alicujus ABG refractus <lb/>B _a_. </s>
  <s xml:id="echoid-s993" xml:space="preserve">Erit ergo PG. </s>
  <s xml:id="echoid-s994" xml:space="preserve">P _a_ :</s>
  <s xml:id="echoid-s995" xml:space="preserve">: (I. </s>
  <s xml:id="echoid-s996" xml:space="preserve">R :</s>
  <s xml:id="echoid-s997" xml:space="preserve">: ) *PH. </s>
  <s xml:id="echoid-s998" xml:space="preserve">PB. </s>
  <s xml:id="echoid-s999" xml:space="preserve">eſt autem PG <lb/>
<anchor type="note" xlink:label="note-0043-05a" xlink:href="note-0043-05"/>
major quàm PH. </s>
  <s xml:id="echoid-s1000" xml:space="preserve">ergo P _a_ major erit quam PB. </s>
  <s xml:id="echoid-s1001" xml:space="preserve">quod planè <lb/>fieri nequit. </s>
  <s xml:id="echoid-s1002" xml:space="preserve">Ergò AB non refringetur in medium ipſi EF ſub-<lb/>jectum.</s>
  <s xml:id="echoid-s1003" xml:space="preserve"/>
</p>
<div xml:id="echoid-div29" type="float" level="2" n="8">
<note position="right" xlink:label="note-0043-05" xlink:href="note-0043-05a" xml:space="preserve">*_Hypotb_.</note>
</div>
<p>
  <s xml:id="echoid-s1004" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s1005" xml:space="preserve">Angulus incidentiæ major ad angulum ſuum refractum ma-<lb/>jorem habet rationem, quam angulus incidentiæ minor ad refra-<lb/>ctum fuum.</s>
  <s xml:id="echoid-s1006" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1007" xml:space="preserve">Erit ſcilicet (in figura numeri Sexti, cujus huc apparatus transfe-<lb/>
<anchor type="note" xlink:label="note-0043-06a" xlink:href="note-0043-06"/>
ratur) ang. </s>
  <s xml:id="echoid-s1008" xml:space="preserve">GBP. </s>
  <s xml:id="echoid-s1009" xml:space="preserve">_a_ BP. </s>
  <s xml:id="echoid-s1010" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s1011" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1012" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1013" xml:space="preserve">HBP. </s>
  <s xml:id="echoid-s1014" xml:space="preserve">δ BP. </s>
  <s xml:id="echoid-s1015" xml:space="preserve">Nam triangùla
<pb o="26" file="0044" n="44" rhead=""/>
GP _a_, HP δ ità diſponantur, ut latera PG, PH ſibi congruant (un-<lb/>de major angulus GP _a_ minorem HP δ comprehendet) tum centro P <lb/>per δ deſcribatur circulus E δ F ipſas PG, P _a_ ſecans punctis F, E; </s>
  <s xml:id="echoid-s1016" xml:space="preserve">item <lb/>connexâ EH, centro H per δ tranſeat circulus HMN ipſas HP, HE <lb/>ſecans punctis N, M; </s>
  <s xml:id="echoid-s1017" xml:space="preserve">denuò connexa E δ cum PG conveniat in L. <lb/></s>
  <s xml:id="echoid-s1018" xml:space="preserve">Eſtque jam ang. </s>
  <s xml:id="echoid-s1019" xml:space="preserve">_a_ P δ. </s>
  <s xml:id="echoid-s1020" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1021" xml:space="preserve">δ PH:</s>
  <s xml:id="echoid-s1022" xml:space="preserve">: ſector EP δ. </s>
  <s xml:id="echoid-s1023" xml:space="preserve">ſector δ PF &amp;</s>
  <s xml:id="echoid-s1024" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1025" xml:space="preserve"><lb/>triang. </s>
  <s xml:id="echoid-s1026" xml:space="preserve">EP δ. </s>
  <s xml:id="echoid-s1027" xml:space="preserve">triang. </s>
  <s xml:id="echoid-s1028" xml:space="preserve">δ PL:</s>
  <s xml:id="echoid-s1029" xml:space="preserve">: Eδ. </s>
  <s xml:id="echoid-s1030" xml:space="preserve">δ L :</s>
  <s xml:id="echoid-s1031" xml:space="preserve">: triang. </s>
  <s xml:id="echoid-s1032" xml:space="preserve">EH δ. </s>
  <s xml:id="echoid-s1033" xml:space="preserve">δ HL &amp;</s>
  <s xml:id="echoid-s1034" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1035" xml:space="preserve"><lb/>ſector MH δ. </s>
  <s xml:id="echoid-s1036" xml:space="preserve">ſector δ HN :</s>
  <s xml:id="echoid-s1037" xml:space="preserve">: ang. </s>
  <s xml:id="echoid-s1038" xml:space="preserve">EH δ. </s>
  <s xml:id="echoid-s1039" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1040" xml:space="preserve">δ HP. </s>
  <s xml:id="echoid-s1041" xml:space="preserve">eſt igi-<lb/>tur ang. </s>
  <s xml:id="echoid-s1042" xml:space="preserve">_a_ P δ. </s>
  <s xml:id="echoid-s1043" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1044" xml:space="preserve">δ PH &amp;</s>
  <s xml:id="echoid-s1045" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1046" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1047" xml:space="preserve">EH δ. </s>
  <s xml:id="echoid-s1048" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1049" xml:space="preserve">δ HP. </s>
  <s xml:id="echoid-s1050" xml:space="preserve">ergóque <lb/>compoſitè ang. </s>
  <s xml:id="echoid-s1051" xml:space="preserve">_a_ PG. </s>
  <s xml:id="echoid-s1052" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1053" xml:space="preserve">δ PH &amp;</s>
  <s xml:id="echoid-s1054" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1055" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1056" xml:space="preserve">EHP. </s>
  <s xml:id="echoid-s1057" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1058" xml:space="preserve">δ HP. </s>
  <s xml:id="echoid-s1059" xml:space="preserve">per-<lb/>mutandóque ang. </s>
  <s xml:id="echoid-s1060" xml:space="preserve">_a_ PG. </s>
  <s xml:id="echoid-s1061" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1062" xml:space="preserve">EHP &amp;</s>
  <s xml:id="echoid-s1063" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1064" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1065" xml:space="preserve">δ PH. </s>
  <s xml:id="echoid-s1066" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1067" xml:space="preserve">δ HP. </s>
  <s xml:id="echoid-s1068" xml:space="preserve">eſt <lb/>autem HP. </s>
  <s xml:id="echoid-s1069" xml:space="preserve">PE :</s>
  <s xml:id="echoid-s1070" xml:space="preserve">: HP. </s>
  <s xml:id="echoid-s1071" xml:space="preserve">P δ :</s>
  <s xml:id="echoid-s1072" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1073" xml:space="preserve">R :</s>
  <s xml:id="echoid-s1074" xml:space="preserve">: GP. </s>
  <s xml:id="echoid-s1075" xml:space="preserve">P _a_. </s>
  <s xml:id="echoid-s1076" xml:space="preserve">adeoque EH ad <lb/>_a_ G parallela; </s>
  <s xml:id="echoid-s1077" xml:space="preserve">vel ang. </s>
  <s xml:id="echoid-s1078" xml:space="preserve">EHP = ang. </s>
  <s xml:id="echoid-s1079" xml:space="preserve">_a_ GP. </s>
  <s xml:id="echoid-s1080" xml:space="preserve">ergò erit ang. </s>
  <s xml:id="echoid-s1081" xml:space="preserve">_a_ PG. </s>
  <s xml:id="echoid-s1082" xml:space="preserve"><lb/>ang. </s>
  <s xml:id="echoid-s1083" xml:space="preserve">_a_ GP &amp;</s>
  <s xml:id="echoid-s1084" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1085" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1086" xml:space="preserve">δ PH. </s>
  <s xml:id="echoid-s1087" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1088" xml:space="preserve">δ HP. </s>
  <s xml:id="echoid-s1089" xml:space="preserve">hoc eſt ang. </s>
  <s xml:id="echoid-s1090" xml:space="preserve">_a_ BG, _a_ BP <lb/>&amp;</s>
  <s xml:id="echoid-s1091" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1092" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1093" xml:space="preserve">δ BH. </s>
  <s xml:id="echoid-s1094" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1095" xml:space="preserve">δ BP. </s>
  <s xml:id="echoid-s1096" xml:space="preserve">vel componendo ang. </s>
  <s xml:id="echoid-s1097" xml:space="preserve">GBP. </s>
  <s xml:id="echoid-s1098" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1099" xml:space="preserve">_a_ BP <lb/>&amp;</s>
  <s xml:id="echoid-s1100" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1101" xml:space="preserve">ang HBP. </s>
  <s xml:id="echoid-s1102" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s1103" xml:space="preserve">δ BP. </s>
  <s xml:id="echoid-s1104" xml:space="preserve">Quod erat demonſtrandum.</s>
  <s xml:id="echoid-s1105" xml:space="preserve"/>
</p>
<div xml:id="echoid-div30" type="float" level="2" n="9">
<note position="right" xlink:label="note-0043-06" xlink:href="note-0043-06a" xml:space="preserve">Fig. 27, 22.</note>
</div>
</div>
<div xml:id="echoid-div32" type="section" level="1" n="12">
<head xml:id="echoid-head15" xml:space="preserve">_Corol_. 1. Ang. _a_ BG. ang. _a_ BP &amp;gt; ang. δ BH. ang. δ BP. <lb/>2. Ang. _a_ BG. ang. PBG &amp;gt; ang. δ BH. PBH.</head>
<p>
  <s xml:id="echoid-s1106" xml:space="preserve">Opportunum eſt hoc Theorema conciliandis cum experientia pro-<lb/>poſitis refractionum legibus. </s>
  <s xml:id="echoid-s1107" xml:space="preserve">Ut demirari ſubeat nuperrimum Opticæ <lb/>ſcriptorem, virum alioqui diffuſè doctum, hujuſmodi ratiocinio leges <lb/>iſtas impugnàſſe: </s>
  <s xml:id="echoid-s1108" xml:space="preserve">“In majoribus tamen angulis inclinationis (Ipſiſ-<lb/>"ſima ſunt ejus verba) falſum eſſe conſtat (principium nempe no-<lb/>"ſtrum;) </s>
  <s xml:id="echoid-s1109" xml:space="preserve">in his enim angulus refractionis major eſt ſubtriplo an-<lb/>"guli inclinationis; </s>
  <s xml:id="echoid-s1110" xml:space="preserve">quod mihi aliiſque ex luculentis experimentis <lb/>"compertum eſt. </s>
  <s xml:id="echoid-s1111" xml:space="preserve">Hæc, inquam, ille ταντοεπ@. </s>
  <s xml:id="echoid-s1112" xml:space="preserve">Quaſi verò dixiſſet; <lb/></s>
  <s xml:id="echoid-s1113" xml:space="preserve">numeri 6 &amp; </s>
  <s xml:id="echoid-s1114" xml:space="preserve">4 ſimul accepti non conficiunt 10, quia numerum effici-<lb/>unt majorem quam 8. </s>
  <s xml:id="echoid-s1115" xml:space="preserve">planè ſimilis eſt diſcurſus; </s>
  <s xml:id="echoid-s1116" xml:space="preserve">non ovum ovo ſi-<lb/>milius. </s>
  <s xml:id="echoid-s1117" xml:space="preserve">Nam in refractionibus ex. </s>
  <s xml:id="echoid-s1118" xml:space="preserve">gr. </s>
  <s xml:id="echoid-s1119" xml:space="preserve">ad vitrum factis ſi ponatur ad <lb/>quamvis inclinationem (puta graduum 15.) </s>
  <s xml:id="echoid-s1120" xml:space="preserve">quòd ſit angulus refra-<lb/>ctionis ſubtriplus anguli inclinationis (quem ille vocat, incidentiæ nos <lb/>angulum appellare ſolemus) neceſſariò, ſicuti modò demonſtratum <lb/>eſt, è principio noſtro conſequetur, quòd ad aliam quamcunque ma-<lb/>jorem inclinationem refractionis angulus major erit ſubtriplo anguli <lb/>inclinationis; </s>
  <s xml:id="echoid-s1121" xml:space="preserve">nominatim acceptâ graduum 30 inclinatione juxta di-<lb/>ctum principium inſtitutus calculus angulum præbebit reſractum <lb/>19.</s>
  <s xml:id="echoid-s1122" xml:space="preserve">24'; </s>
  <s xml:id="echoid-s1123" xml:space="preserve">angulúmque proinde refractionis 10. </s>
  <s xml:id="echoid-s1124" xml:space="preserve">36', qui 30 graduum <lb/>trientem exuperat. </s>
  <s xml:id="echoid-s1125" xml:space="preserve">Quare cùm Clariſſimus vir Hypotheſin hanc (à
<pb o="27" file="0045" n="45" rhead=""/>
_Carteſio_ quidem primò repertam, ſed ab aliis pleriſque recentioribus <lb/>opticis _Merſenno. </s>
  <s xml:id="echoid-s1126" xml:space="preserve">Herigonio, Hobbio, Maignano_, quin &amp; </s>
  <s xml:id="echoid-s1127" xml:space="preserve">ipſo ejus <lb/>conſodale doctiſſimo _Ricciolo_ ſuſceptam &amp; </s>
  <s xml:id="echoid-s1128" xml:space="preserve">approbatam; </s>
  <s xml:id="echoid-s1129" xml:space="preserve">quam &amp; </s>
  <s xml:id="echoid-s1130" xml:space="preserve">cer-<lb/>tè hujus Scientiæ non parùm intereſt veram deprehendi) labefactatum <lb/>iret; </s>
  <s xml:id="echoid-s1131" xml:space="preserve">eam potiùs imprudens experientiæ Suffragio communivit. </s>
  <s xml:id="echoid-s1132" xml:space="preserve">Qui-<lb/>nimò ſi quid inſit huic principio vitii, illud potiùs erit, quod in maxi-<lb/>mis inclinationibus refractionis angulos exhibet apparentibus ali-<lb/>quantillo majores; </s>
  <s xml:id="echoid-s1133" xml:space="preserve">quæ tamen diſcrepantia num ipſius legis hujus, an <lb/>experimentorum defectui, vel accidentariis quíbuſdam intervenienti-<lb/>bus cauſis adſcribi debeat, haud facilè pronunciaverim. </s>
  <s xml:id="echoid-s1134" xml:space="preserve">Nec enim <lb/>fortaſſis cognata reflectionis lex, a nemine non admiſſa, experimen-<lb/>tis omnibus præciſè reſponderet. </s>
  <s xml:id="echoid-s1135" xml:space="preserve">Nobis ſuſficiet quòd in reliquis in-<lb/>clinationibus, mediis præſertim, dicta lex experientiæ, quam præfe-<lb/>runt authores, perquam conſentanea reperitur; </s>
  <s xml:id="echoid-s1136" xml:space="preserve">addo, quòd ab ea de-<lb/>ductæ concluſiones cum experientia mirè conſpirant; </s>
  <s xml:id="echoid-s1137" xml:space="preserve">nec ab ea quòd <lb/>animadvertere potuerim, unquam diſcordant. </s>
  <s xml:id="echoid-s1138" xml:space="preserve">Eam proinde (cùm alia <lb/>probabilis haud ſuppetat, Geometricis, ratiociniis præſternenda) non <lb/>verebimur ubivis ut ratam ſumere, ac adhibere; </s>
  <s xml:id="echoid-s1139" xml:space="preserve">ſatis certi (apud nos <lb/>ſaltem) in elicitis ab ea concluſionibus haud omnino quicquam nota-<lb/>bilis erroris emerſurum.</s>
  <s xml:id="echoid-s1140" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1141" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s1142" xml:space="preserve">Obiter hîc &amp; </s>
  <s xml:id="echoid-s1143" xml:space="preserve">παρεπβαπκ\~ες problemation quoddam interſeram <lb/>
<anchor type="note" xlink:label="note-0045-01a" xlink:href="note-0045-01"/>
(quia Schema num. </s>
  <s xml:id="echoid-s1144" xml:space="preserve">6. </s>
  <s xml:id="echoid-s1145" xml:space="preserve">ſuperius ei gratis inſerviet, ejuſque conſtructio <lb/>è ſuperiore conſtructione derivatur.) </s>
  <s xml:id="echoid-s1146" xml:space="preserve">Per datum in refringente pun-<lb/>ctum (B) incidentem ducere, cui datus conveniat angulus refractionis. <lb/></s>
  <s xml:id="echoid-s1147" xml:space="preserve">Ducatur BP refringenti perpendicularis (hanc autem duci poſſe ſup-<lb/>ponimus, aut poſtulamus) &amp; </s>
  <s xml:id="echoid-s1148" xml:space="preserve">ad diametrum BP conſtruatur ſemicir-<lb/>culus; </s>
  <s xml:id="echoid-s1149" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1150" xml:space="preserve">ſit utcunque PG. </s>
  <s xml:id="echoid-s1151" xml:space="preserve">P _a_ :</s>
  <s xml:id="echoid-s1152" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1153" xml:space="preserve">R (pro PG verò præſtat ipſam <lb/>diametrum PB accipere) ſumatúrque Arcus GK ſubtendens angulum <lb/>parem dato. </s>
  <s xml:id="echoid-s1154" xml:space="preserve">Fiat autem PX = P _a_. </s>
  <s xml:id="echoid-s1155" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1156" xml:space="preserve">per G, X ducta recta circulo <lb/>occurrat in δ. </s>
  <s xml:id="echoid-s1157" xml:space="preserve">demum accipiatur Arcus δ H = GK, erit ductæ BH <lb/>refractus B δ (uti præcedentem diſcurſum invertendo non difficilè col-<lb/>ligitur) adeóque liquet factum eſſe quod erat propoſitum. </s>
  <s xml:id="echoid-s1158" xml:space="preserve">Hoc præter <lb/>ordinem; </s>
  <s xml:id="echoid-s1159" xml:space="preserve">ergo perfunctoriè.</s>
  <s xml:id="echoid-s1160" xml:space="preserve"/>
</p>
<div xml:id="echoid-div32" type="float" level="2" n="1">
<note position="right" xlink:label="note-0045-01" xlink:href="note-0045-01a" xml:space="preserve">Fig 21.</note>
</div>
<p>
  <s xml:id="echoid-s1161" xml:space="preserve">X. </s>
  <s xml:id="echoid-s1162" xml:space="preserve">Cujuſcunque generis lineæ RBS incidat radius MNO ad N, <lb/>
<anchor type="note" xlink:label="note-0045-02a" xlink:href="note-0045-02"/>
ſitque dictæ lineæ perpendicularis recta NC; </s>
  <s xml:id="echoid-s1163" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1164" xml:space="preserve">in hac utcunque <lb/>ſumpto puncto C, per hoc tranſeat incidenti parallela CB; </s>
  <s xml:id="echoid-s1165" xml:space="preserve">quâcum <lb/>conveniat ipſius MO inflexus GNK; </s>
  <s xml:id="echoid-s1166" xml:space="preserve">erit in reflectione KN = KC; <lb/></s>
  <s xml:id="echoid-s1167" xml:space="preserve">in refractione verò KN. </s>
  <s xml:id="echoid-s1168" xml:space="preserve">KC :</s>
  <s xml:id="echoid-s1169" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1170" xml:space="preserve">R. </s>
  <s xml:id="echoid-s1171" xml:space="preserve">(vel item, ſi in ipſo inflexo
<pb o="28" file="0046" n="46" rhead=""/>
ſumatur utcunque punctum K; </s>
  <s xml:id="echoid-s1172" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1173" xml:space="preserve">ab eo ducta KL ad perpendicularem <lb/>CN parallela cum incidente conveniet ad L, erit illic KN = NL; </s>
  <s xml:id="echoid-s1174" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1175" xml:space="preserve"><lb/>hìc KN. </s>
  <s xml:id="echoid-s1176" xml:space="preserve">NL :</s>
  <s xml:id="echoid-s1177" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1178" xml:space="preserve">R.)</s>
  <s xml:id="echoid-s1179" xml:space="preserve"/>
</p>
<div xml:id="echoid-div33" type="float" level="2" n="2">
<note position="right" xlink:label="note-0045-02" xlink:href="note-0045-02a" xml:space="preserve">Fig. 24, 25.</note>
</div>
<p>
  <s xml:id="echoid-s1180" xml:space="preserve">Nam 1. </s>
  <s xml:id="echoid-s1181" xml:space="preserve">in reflectione; </s>
  <s xml:id="echoid-s1182" xml:space="preserve">quoniam ang. </s>
  <s xml:id="echoid-s1183" xml:space="preserve">ONC = KNC (ex lege <lb/>
<anchor type="note" xlink:label="note-0046-01a" xlink:href="note-0046-01"/>
reflectionis.) </s>
  <s xml:id="echoid-s1184" xml:space="preserve">Etang. </s>
  <s xml:id="echoid-s1185" xml:space="preserve">ONC = KCN (ex Hypotheſi quòd ON, <lb/>CB parallelæ ſunt ) erit ang. </s>
  <s xml:id="echoid-s1186" xml:space="preserve">KCN = ang. </s>
  <s xml:id="echoid-s1187" xml:space="preserve">KNC. </s>
  <s xml:id="echoid-s1188" xml:space="preserve">adeóque KN <lb/> = KC = NL: </s>
  <s xml:id="echoid-s1189" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s1190" xml:space="preserve">E. </s>
  <s xml:id="echoid-s1191" xml:space="preserve">D.</s>
  <s xml:id="echoid-s1192" xml:space="preserve"/>
</p>
<div xml:id="echoid-div34" type="float" level="2" n="3">
<note position="left" xlink:label="note-0046-01" xlink:href="note-0046-01a" xml:space="preserve">Fig. 24, 25.</note>
</div>
<p>
  <s xml:id="echoid-s1193" xml:space="preserve">2 In refractione; </s>
  <s xml:id="echoid-s1194" xml:space="preserve">ducantur CE ad NO, &amp; </s>
  <s xml:id="echoid-s1195" xml:space="preserve">CF ad NK perpen-<lb/>diculares (unde liquet puncta E, F exiſtere in circulo ſuper diametrum <lb/>CN deſcripto) quare, connexâ EF; </s>
  <s xml:id="echoid-s1196" xml:space="preserve">erunt anguli CEF = ang. <lb/></s>
  <s xml:id="echoid-s1197" xml:space="preserve">FNC (eidem inſiſtentes peripheriæ FC) æquales. </s>
  <s xml:id="echoid-s1198" xml:space="preserve">Item propterea <lb/>eſt ang. </s>
  <s xml:id="echoid-s1199" xml:space="preserve">ECF = ang. </s>
  <s xml:id="echoid-s1200" xml:space="preserve">FNE = ang. </s>
  <s xml:id="echoid-s1201" xml:space="preserve">NKC. </s>
  <s xml:id="echoid-s1202" xml:space="preserve">quare triangula ECF, <lb/>NKC ſunt æquiangula ſibi mutuo; </s>
  <s xml:id="echoid-s1203" xml:space="preserve">quamobrem eſt CE. </s>
  <s xml:id="echoid-s1204" xml:space="preserve">CF:</s>
  <s xml:id="echoid-s1205" xml:space="preserve">: <lb/>KN. </s>
  <s xml:id="echoid-s1206" xml:space="preserve">KC. </s>
  <s xml:id="echoid-s1207" xml:space="preserve">atqui (juxta legem refractionis) eſt CE. </s>
  <s xml:id="echoid-s1208" xml:space="preserve">CF:</s>
  <s xml:id="echoid-s1209" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1210" xml:space="preserve">R. </s>
  <s xml:id="echoid-s1211" xml:space="preserve"><lb/>qua propter erit, KN .</s>
  <s xml:id="echoid-s1212" xml:space="preserve">. KC :</s>
  <s xml:id="echoid-s1213" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1214" xml:space="preserve">R : </s>
  <s xml:id="echoid-s1215" xml:space="preserve">vel KN. </s>
  <s xml:id="echoid-s1216" xml:space="preserve">NL :</s>
  <s xml:id="echoid-s1217" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1218" xml:space="preserve">R: </s>
  <s xml:id="echoid-s1219" xml:space="preserve"><lb/>Q. </s>
  <s xml:id="echoid-s1220" xml:space="preserve">E. </s>
  <s xml:id="echoid-s1221" xml:space="preserve">D.</s>
  <s xml:id="echoid-s1222" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1223" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s1224" xml:space="preserve">Quod ſi per N ducatur tangens UT; </s>
  <s xml:id="echoid-s1225" xml:space="preserve">erit (in reflectione) eti-<lb/>am KT = KN; </s>
  <s xml:id="echoid-s1226" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1227" xml:space="preserve">NT angulum MNK biſecabit. </s>
  <s xml:id="echoid-s1228" xml:space="preserve">In refracti-<lb/>one verò erit KT ad KN, ut co-ſinus anguli refracti, ad coſinum an-<lb/>guli incidentiæ. </s>
  <s xml:id="echoid-s1229" xml:space="preserve">Quæ ſaltem ad noto, ceu Lemmatica.</s>
  <s xml:id="echoid-s1230" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1231" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s1232" xml:space="preserve">Exhis facilè deducantur Conicarum Sectionum circa radiorum <lb/>inflectionem ſatis jam pervulgatæ proprietates; </s>
  <s xml:id="echoid-s1233" xml:space="preserve">at quæ fortaſsè per <lb/>nimias ambages.</s>
  <s xml:id="echoid-s1234" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1235" xml:space="preserve">1. </s>
  <s xml:id="echoid-s1236" xml:space="preserve">Demonſtratæ proſtant. </s>
  <s xml:id="echoid-s1237" xml:space="preserve">Ut in parabola (puta RBS, cujus axis <lb/>BC) incidat MNO axi BC parallelus; </s>
  <s xml:id="echoid-s1238" xml:space="preserve">ejuſque reflexus ſit NK; <lb/></s>
  <s xml:id="echoid-s1239" xml:space="preserve">crit igitur (ex oſtenſis) KN = KC. </s>
  <s xml:id="echoid-s1240" xml:space="preserve">at ſi punctum K ponatur umbi-<lb/>licus parabolæ; </s>
  <s xml:id="echoid-s1241" xml:space="preserve">erit etiam indè (juxta notiſſimam hujuſce curvæ pro-<lb/>prietatem) KN = KC. </s>
  <s xml:id="echoid-s1242" xml:space="preserve">quare paralleli radii reflexus neceſſariò <lb/>per umbilicum tranſibit; </s>
  <s xml:id="echoid-s1243" xml:space="preserve">qui proptereà non immeritò quoque _focus_ <lb/>appèllatur.</s>
  <s xml:id="echoid-s1244" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1245" xml:space="preserve">2. </s>
  <s xml:id="echoid-s1246" xml:space="preserve">Item _in ellipſe_, cujus axis BD, foci H, K, ſi ad quodvis curvæ <lb/>
<anchor type="note" xlink:label="note-0046-02a" xlink:href="note-0046-02"/>
punctum N à focis ducantur rectæ HN, KN; </s>
  <s xml:id="echoid-s1247" xml:space="preserve">ſatis celebre eſt, quòd <lb/>perpendicularis CN angulum HNK biſecabit. </s>
  <s xml:id="echoid-s1248" xml:space="preserve">Unde NH. </s>
  <s xml:id="echoid-s1249" xml:space="preserve">NK:</s>
  <s xml:id="echoid-s1250" xml:space="preserve">: <lb/>HC. </s>
  <s xml:id="echoid-s1251" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s1252" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1253" xml:space="preserve">componendo NH + NK. </s>
  <s xml:id="echoid-s1254" xml:space="preserve">NK :</s>
  <s xml:id="echoid-s1255" xml:space="preserve">: HK. </s>
  <s xml:id="echoid-s1256" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s1257" xml:space="preserve">vel BD. <lb/></s>
  <s xml:id="echoid-s1258" xml:space="preserve">NK :</s>
  <s xml:id="echoid-s1259" xml:space="preserve">: HK. </s>
  <s xml:id="echoid-s1260" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s1261" xml:space="preserve">vel permutando BD. </s>
  <s xml:id="echoid-s1262" xml:space="preserve">HK :</s>
  <s xml:id="echoid-s1263" xml:space="preserve">: NK. </s>
  <s xml:id="echoid-s1264" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s1265" xml:space="preserve">quare <lb/>ſi talis@fuerit ellipſis, ut ſit BD. </s>
  <s xml:id="echoid-s1266" xml:space="preserve">HK :</s>
  <s xml:id="echoid-s1267" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1268" xml:space="preserve">R. </s>
  <s xml:id="echoid-s1269" xml:space="preserve">etiam erit NK. </s>
  <s xml:id="echoid-s1270" xml:space="preserve">CK:</s>
  <s xml:id="echoid-s1271" xml:space="preserve">: <lb/>I. </s>
  <s xml:id="echoid-s1272" xml:space="preserve">R. </s>
  <s xml:id="echoid-s1273" xml:space="preserve">verum ſi incidens MN ad BD parallelus refringatur in NK; </s>
  <s xml:id="echoid-s1274" xml:space="preserve"><lb/>erit (juxta mox oſtenſa) etiam NK. </s>
  <s xml:id="echoid-s1275" xml:space="preserve">CK:</s>
  <s xml:id="echoid-s1276" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1277" xml:space="preserve">R. </s>
  <s xml:id="echoid-s1278" xml:space="preserve">patet itaque quòd <lb/>ipſius MN refractus per focum K tranſibit, Quid plura?</s>
  <s xml:id="echoid-s1279" xml:space="preserve"/>
</p>
<div xml:id="echoid-div35" type="float" level="2" n="4">
<note position="left" xlink:label="note-0046-02" xlink:href="note-0046-02a" xml:space="preserve">Fig. 26.</note>
</div>
<pb o="29" file="0047" n="47" rhead=""/>
<p>
  <s xml:id="echoid-s1280" xml:space="preserve">3. </s>
  <s xml:id="echoid-s1281" xml:space="preserve">Non abſimiliter in _Hyperbola_, (cujus itidem axis BD, foci H, K; <lb/></s>
  <s xml:id="echoid-s1282" xml:space="preserve">reliquiſque velut anteà præparatis) oſtendetur fore perpetuò NK. </s>
  <s xml:id="echoid-s1283" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0047-01a" xlink:href="note-0047-01"/>
CK :</s>
  <s xml:id="echoid-s1284" xml:space="preserve">: BD. </s>
  <s xml:id="echoid-s1285" xml:space="preserve">HK. </s>
  <s xml:id="echoid-s1286" xml:space="preserve">unde ſi fuerit (ex _Hyperbolæ_ conſtructione) BD. <lb/></s>
  <s xml:id="echoid-s1287" xml:space="preserve">HK :</s>
  <s xml:id="echoid-s1288" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1289" xml:space="preserve">R. </s>
  <s xml:id="echoid-s1290" xml:space="preserve">erit etiam NK. </s>
  <s xml:id="echoid-s1291" xml:space="preserve">CK :</s>
  <s xml:id="echoid-s1292" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1293" xml:space="preserve">R. </s>
  <s xml:id="echoid-s1294" xml:space="preserve">quòd ſi radius MN ad <lb/>CB parallelus refringatur in NK; </s>
  <s xml:id="echoid-s1295" xml:space="preserve">hoc idem accidet, ut nempe ſit <lb/>NK. </s>
  <s xml:id="echoid-s1296" xml:space="preserve">CK :</s>
  <s xml:id="echoid-s1297" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1298" xml:space="preserve">R. </s>
  <s xml:id="echoid-s1299" xml:space="preserve">quare radii MN refractus per _Hyperbolæ_ focum <lb/>tranſibit.</s>
  <s xml:id="echoid-s1300" xml:space="preserve"/>
</p>
<div xml:id="echoid-div36" type="float" level="2" n="5">
<note position="right" xlink:label="note-0047-01" xlink:href="note-0047-01a" xml:space="preserve">Fig. 27.</note>
</div>
<p>
  <s xml:id="echoid-s1301" xml:space="preserve">4. </s>
  <s xml:id="echoid-s1302" xml:space="preserve">Quòd verò ab _Ellipſis_ aut _Hyperbolæ_ cujuſvis focorum alterutro <lb/>quilibet curvæ incidens radieus in alterum reflectatur, admodum facilè <lb/>diluceſcit. </s>
  <s xml:id="echoid-s1303" xml:space="preserve">Nam in ellipſe, perpendicularis NC, in _Hyperbolæ_, tan-<lb/>gens. </s>
  <s xml:id="echoid-s1304" xml:space="preserve">NT biſecat angulum HNK. </s>
  <s xml:id="echoid-s1305" xml:space="preserve">unde patet propoſitum, Hæc <lb/>extra noſtras oleas poſita curſim &amp; </s>
  <s xml:id="echoid-s1306" xml:space="preserve">leviſſimè perſtringo; </s>
  <s xml:id="echoid-s1307" xml:space="preserve">nec tamen <lb/>ut eò multa putem deſiderari.</s>
  <s xml:id="echoid-s1308" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1309" xml:space="preserve">Revertamur in orbitani; </s>
  <s xml:id="echoid-s1310" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1311" xml:space="preserve">quidem derelictis his generaliſſimis, <lb/>ac abſtractiſſimis, lemmatum vicem obituris, ad particularia deſcenda-<lb/>mus. </s>
  <s xml:id="echoid-s1312" xml:space="preserve">Ad planas verò ſuperficies (vel earum loco propter inſinuatam <lb/>antehac cauſam ſubrogatas lineas rectas) inflexis obtingentia radiis pri-<lb/>mò contemplemur. </s>
  <s xml:id="echoid-s1313" xml:space="preserve">Etiam quoad has Catoptricis primum, utpote fa-<lb/>cillimis, breviſſimè defungemur.</s>
  <s xml:id="echoid-s1314" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1315" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s1316" xml:space="preserve">1. </s>
  <s xml:id="echoid-s1317" xml:space="preserve">Parallelorum ſibi radiorum (AB, MN) rectæ (EF) in-<lb/>
<anchor type="note" xlink:label="note-0047-02a" xlink:href="note-0047-02"/>
cidentium reflexi (B _α_, Nμ) ſunt etiam ſibi paralleli.</s>
  <s xml:id="echoid-s1318" xml:space="preserve"/>
</p>
<div xml:id="echoid-div37" type="float" level="2" n="6">
<note position="right" xlink:label="note-0047-02" xlink:href="note-0047-02a" xml:space="preserve">Fig. 28.</note>
</div>
<p>
  <s xml:id="echoid-s1319" xml:space="preserve">Nam quoniam AB, MN ex hypotheſi ſunt pàralleli, erunt anguli <lb/>ABE, MNE pares. </s>
  <s xml:id="echoid-s1320" xml:space="preserve">Ergò ſunt anguli _α_ BF, μ NF etiam pares. <lb/></s>
  <s xml:id="echoid-s1321" xml:space="preserve">Quare rectæ _α_ B, μ N ſunt parallelæ.</s>
  <s xml:id="echoid-s1322" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1323" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s1324" xml:space="preserve">2. </s>
  <s xml:id="echoid-s1325" xml:space="preserve">Sit recta ABZ rectæ reflectenti EF perpendicularis; </s>
  <s xml:id="echoid-s1326" xml:space="preserve">cum <lb/>hacverò promanantis ab A cujuſvis radii AN reflexus _α_ N. </s>
  <s xml:id="echoid-s1327" xml:space="preserve">conveniat <lb/>in Z; </s>
  <s xml:id="echoid-s1328" xml:space="preserve">dico forè BZ = AB. </s>
  <s xml:id="echoid-s1329" xml:space="preserve">Nam ang. </s>
  <s xml:id="echoid-s1330" xml:space="preserve">ANB = ang. </s>
  <s xml:id="echoid-s1331" xml:space="preserve">_α_ NF = <lb/>ZNB. </s>
  <s xml:id="echoid-s1332" xml:space="preserve">quare liquet triangula BNA, BNZ ſibi mutuò æquilatera <lb/>fore; </s>
  <s xml:id="echoid-s1333" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1334" xml:space="preserve">eſſe AB = BZ: </s>
  <s xml:id="echoid-s1335" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s1336" xml:space="preserve">E. </s>
  <s xml:id="echoid-s1337" xml:space="preserve">D.</s>
  <s xml:id="echoid-s1338" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1339" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s1340" xml:space="preserve">3. </s>
  <s xml:id="echoid-s1341" xml:space="preserve">Hinc, omnes ab uno puncto, divergentium tanquam ab altero <lb/>
<anchor type="note" xlink:label="note-0047-03a" xlink:href="note-0047-03"/>
quodam uno prodeuntes.</s>
  <s xml:id="echoid-s1342" xml:space="preserve"/>
</p>
<div xml:id="echoid-div38" type="float" level="2" n="7">
<note position="right" xlink:label="note-0047-03" xlink:href="note-0047-03a" xml:space="preserve">Fig. 29.</note>
</div>
<p>
  <s xml:id="echoid-s1343" xml:space="preserve">Quoad punctum longè diſſitum (ſuo parallelos ad ſenſum radios eja-<lb/>@ulante) patet è penultima. </s>
  <s xml:id="echoid-s1344" xml:space="preserve">Quòad punctum è ſenſibiliter finita di-<lb/>ſtantia radians, ex ultima patet, quòd omnium ab A divergentium ra-<lb/>diorum reflexi protracti concurrunt in Z; </s>
  <s xml:id="echoid-s1345" xml:space="preserve">adeóque videbuntur ab eo <lb/>promanare.</s>
  <s xml:id="echoid-s1346" xml:space="preserve"/>
</p>
<pb o="30" file="0048" n="48" rhead=""/>
<p>
  <s xml:id="echoid-s1347" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s1348" xml:space="preserve">Hinc punctum Z erit ipſius A (reſpectu oculi uspiam conſti-<lb/>tuti) imago perfectiſſima. </s>
  <s xml:id="echoid-s1349" xml:space="preserve">Siquidem imaginis vocabulo nil aliud in-<lb/>telligo, quàm locum à quo plures radii (quot ſcilicet afficiendo viſui <lb/>ſufficiunt) ſimiliter divergere, ſeu dimanare videntur, atque cùm à <lb/>primariis objectis diffunduntur. </s>
  <s xml:id="echoid-s1350" xml:space="preserve">Proinde cujuſvis hoc modo radiantis <lb/>objecti locus apparens, vel imago facilimè determinatur.</s>
  <s xml:id="echoid-s1351" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1352" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s1353" xml:space="preserve">Exhinc etiam eâdem operâ, viſùs imaginem adſpectantis axis, <lb/>ſeu reflexus principalis (iſte nimirum qui per oculi centrum (puta O) <lb/>tranſit,) &amp; </s>
  <s xml:id="echoid-s1354" xml:space="preserve">reflectionis (quod vocant) punctum determinantur. </s>
  <s xml:id="echoid-s1355" xml:space="preserve">Con-<lb/>nexa nempe recta OZ@erit axis iſte; </s>
  <s xml:id="echoid-s1356" xml:space="preserve">nec non ejus cum EF interſectio <lb/>N, punctum reflectionis.</s>
  <s xml:id="echoid-s1357" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1358" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s1359" xml:space="preserve">Quoad hoc reflectionis punctum unicam ſubjiciemus anno-<lb/>tatiunculam. </s>
  <s xml:id="echoid-s1360" xml:space="preserve">Radiante puncto A, &amp; </s>
  <s xml:id="echoid-s1361" xml:space="preserve">oculi centro O fixis manentibus <lb/>recta Catoptrica EF ponatur rectæ cuidam OP parallela, ſed alioquin <lb/>ſitu indeterminata; </s>
  <s xml:id="echoid-s1362" xml:space="preserve">erunt omnia reflectionis puncta in _Hyperbolæ_. <lb/></s>
  <s xml:id="echoid-s1363" xml:space="preserve">Sit, inquam, AP ad OP perpendicularis, &amp; </s>
  <s xml:id="echoid-s1364" xml:space="preserve">biſecentur AP in X, at-<lb/>que PO in Y; </s>
  <s xml:id="echoid-s1365" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1366" xml:space="preserve">per X ducatur XG ad PO parallela, item per Y <lb/>
<anchor type="note" xlink:label="note-0048-01a" xlink:href="note-0048-01"/>
ducatur YH ad AP parallela; </s>
  <s xml:id="echoid-s1367" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1368" xml:space="preserve">XG, YH concurrant in C; </s>
  <s xml:id="echoid-s1369" xml:space="preserve">tum <lb/>Aſymptotis CG, CH per ipſum O deſcripta concipiatur _Hyperbole_ <lb/>ROS; </s>
  <s xml:id="echoid-s1370" xml:space="preserve">hæc per omnia reflectionum dictarum puncta tranſibit. </s>
  <s xml:id="echoid-s1371" xml:space="preserve">Nam <lb/>utcunque ducta EF ad PO parallela _Hyperbolæ_ ROS occurrat ad N; <lb/></s>
  <s xml:id="echoid-s1372" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1373" xml:space="preserve">ducantur rectæ AN, ON; </s>
  <s xml:id="echoid-s1374" xml:space="preserve">dico angulnm ANE angulo ONF <lb/>æquari. </s>
  <s xml:id="echoid-s1375" xml:space="preserve">Secet enim AP ipſam EF in B; </s>
  <s xml:id="echoid-s1376" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1377" xml:space="preserve">ducatur OQ ad AB <lb/>parallela. </s>
  <s xml:id="echoid-s1378" xml:space="preserve">Et, ex _Hyperbolæ_ natura, eſt CD. </s>
  <s xml:id="echoid-s1379" xml:space="preserve">CY :</s>
  <s xml:id="echoid-s1380" xml:space="preserve">: YODN. </s>
  <s xml:id="echoid-s1381" xml:space="preserve"><lb/>quare dividendo erit YD. </s>
  <s xml:id="echoid-s1382" xml:space="preserve">CY :</s>
  <s xml:id="echoid-s1383" xml:space="preserve">: YO-DNDN; </s>
  <s xml:id="echoid-s1384" xml:space="preserve">hoc eſt OQ. </s>
  <s xml:id="echoid-s1385" xml:space="preserve"><lb/>CY :</s>
  <s xml:id="echoid-s1386" xml:space="preserve">: NQDN. </s>
  <s xml:id="echoid-s1387" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1388" xml:space="preserve">permutatim OQNQ :</s>
  <s xml:id="echoid-s1389" xml:space="preserve">: CY. </s>
  <s xml:id="echoid-s1390" xml:space="preserve">DN. </s>
  <s xml:id="echoid-s1391" xml:space="preserve">item <lb/>rurſus ob CD. </s>
  <s xml:id="echoid-s1392" xml:space="preserve">CY :</s>
  <s xml:id="echoid-s1393" xml:space="preserve">: YO. </s>
  <s xml:id="echoid-s1394" xml:space="preserve">DN. </s>
  <s xml:id="echoid-s1395" xml:space="preserve">erit componendo CD + CY. </s>
  <s xml:id="echoid-s1396" xml:space="preserve"><lb/>CY :</s>
  <s xml:id="echoid-s1397" xml:space="preserve">: YO + DN. </s>
  <s xml:id="echoid-s1398" xml:space="preserve">DN. </s>
  <s xml:id="echoid-s1399" xml:space="preserve">hoc eſt AB. </s>
  <s xml:id="echoid-s1400" xml:space="preserve">CY :</s>
  <s xml:id="echoid-s1401" xml:space="preserve">: BN. </s>
  <s xml:id="echoid-s1402" xml:space="preserve">DN. </s>
  <s xml:id="echoid-s1403" xml:space="preserve">vel <lb/>permutando AB. </s>
  <s xml:id="echoid-s1404" xml:space="preserve">BN :</s>
  <s xml:id="echoid-s1405" xml:space="preserve">: CY. </s>
  <s xml:id="echoid-s1406" xml:space="preserve">DN. </s>
  <s xml:id="echoid-s1407" xml:space="preserve">quare eſt OQ. </s>
  <s xml:id="echoid-s1408" xml:space="preserve">NQ :</s>
  <s xml:id="echoid-s1409" xml:space="preserve">: AB. </s>
  <s xml:id="echoid-s1410" xml:space="preserve"><lb/>BN. </s>
  <s xml:id="echoid-s1411" xml:space="preserve">ergò rectangula triangula, OQN, ABN ſimilia ſunt; </s>
  <s xml:id="echoid-s1412" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1413" xml:space="preserve">patet <lb/>angulum ONQ angulo ANB æquari: </s>
  <s xml:id="echoid-s1414" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s1415" xml:space="preserve">E. </s>
  <s xml:id="echoid-s1416" xml:space="preserve">D.</s>
  <s xml:id="echoid-s1417" xml:space="preserve"/>
</p>
<div xml:id="echoid-div39" type="float" level="2" n="8">
<note position="left" xlink:label="note-0048-01" xlink:href="note-0048-01a" xml:space="preserve">Fig. 30.</note>
</div>
<p>
  <s xml:id="echoid-s1418" xml:space="preserve">Mereri ſaltem vel _Hyperbolæ gratiâ_ videbatur hæc ejuſce proprietas <lb/>adnotari; </s>
  <s xml:id="echoid-s1419" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s1420" xml:space="preserve">Analogiæ cauſà verſus ea quæ ſequentur. </s>
  <s xml:id="echoid-s1421" xml:space="preserve">Neque <lb/>de reflectionibus ad plana quicquam prætereà. </s>
  <s xml:id="echoid-s1422" xml:space="preserve">Ad refractiones <lb/>tranſeo.</s>
  <s xml:id="echoid-s1423" xml:space="preserve"/>
</p>
<pb o="31" file="0049" n="49"/>
</div>
<div xml:id="echoid-div41" type="section" level="1" n="13">
<head xml:id="echoid-head16" xml:space="preserve"><emph style="sc">Lect</emph>. IV.</head>
<p>
  <s xml:id="echoid-s1424" xml:space="preserve">I. </s>
  <s xml:id="echoid-s1425" xml:space="preserve">AD ea jam accedimus quæ radiis obveniunt ad planam ſuperfici-<lb/>em, vel ad rectam lineam, refractis. </s>
  <s xml:id="echoid-s1426" xml:space="preserve">Quod argumentum eo <lb/>diligentiùs proſequemur, quia nondum pro merito ſuo videtur ſatis ex-<lb/>cultum; </s>
  <s xml:id="echoid-s1427" xml:space="preserve">ut &amp; </s>
  <s xml:id="echoid-s1428" xml:space="preserve">quoniam in eo tractando methodum præſtituemus no-<lb/>bis, &amp; </s>
  <s xml:id="echoid-s1429" xml:space="preserve">quaſi normam in ſequentibus obſervandam. </s>
  <s xml:id="echoid-s1430" xml:space="preserve">Ad rem.</s>
  <s xml:id="echoid-s1431" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1432" xml:space="preserve">II. </s>
  <s xml:id="echoid-s1433" xml:space="preserve">Parallelorum rectæ lineæ (EF) incidentium radiorum (AB, <lb/>
<anchor type="note" xlink:label="note-0049-01a" xlink:href="note-0049-01"/>
MN) refracti (B _a_, N μ) ſunt etiam ſibi paralleli. </s>
  <s xml:id="echoid-s1434" xml:space="preserve">Nam quoniam <lb/>AB, MN ſunt, ex hypotheſi, paralleli, erunt anguli ABE, MNE <lb/>pares. </s>
  <s xml:id="echoid-s1435" xml:space="preserve">Itaque refractos habent angulos pares; </s>
  <s xml:id="echoid-s1436" xml:space="preserve">horúmque comple-<lb/>menta (ſcilicet anguli _a_ BF, μ NF) æquantur, quare liquet refractos <lb/>B _a_, N μ ſibi parallelos eſſe.</s>
  <s xml:id="echoid-s1437" xml:space="preserve"/>
</p>
<div xml:id="echoid-div41" type="float" level="2" n="1">
<note position="right" xlink:label="note-0049-01" xlink:href="note-0049-01a" xml:space="preserve">Fig. 31.</note>
</div>
<p>
  <s xml:id="echoid-s1438" xml:space="preserve">III. </s>
  <s xml:id="echoid-s1439" xml:space="preserve">Hinc infinitè diſtantis, hoc eſt parallelos radios emittentis (in-<lb/>finitam ad ſenſum diſtantiam intelligo, qualis eſt quoad hoc ſtellæ cu-<lb/>juſpiam) puncti locus apparens, aut imago per hujuſmodi reſractio-<lb/>nem eſſecta infinitè quoque diſtat; </s>
  <s xml:id="echoid-s1440" xml:space="preserve">quippe cùm hæc etiam per radios <lb/>parallelos adſpectetur. </s>
  <s xml:id="echoid-s1441" xml:space="preserve">Itaque ſitus ejus reſpectu visûs ubivis poſiti fa-<lb/>cilè determinatur. </s>
  <s xml:id="echoid-s1442" xml:space="preserve">Sit oculi puta centrum O; </s>
  <s xml:id="echoid-s1443" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1444" xml:space="preserve">A punctum radians <lb/>immenſè diſſitum; </s>
  <s xml:id="echoid-s1445" xml:space="preserve">connexáque AO refringentem EF ſecet in G; <lb/></s>
  <s xml:id="echoid-s1446" xml:space="preserve">ſitque radii AG reſractus G _a_;</s>
  <s xml:id="echoid-s1447" xml:space="preserve">; per O verò ducatur OBZ ad _a_ G <lb/>
<anchor type="note" xlink:label="note-0049-02a" xlink:href="note-0049-02"/>
parallela; </s>
  <s xml:id="echoid-s1448" xml:space="preserve">in hac ad infinitum protenſa (velut ad Z) apparebit pun-<lb/>ctum A. </s>
  <s xml:id="echoid-s1449" xml:space="preserve">Cùm enim radii AG, AB ſint (ad ſenſum) paralleli, eti-<lb/>am ipſorum refracti erunt paralleli. </s>
  <s xml:id="echoid-s1450" xml:space="preserve">Quare cùm G _a_ ſit refractus ipſius <lb/>AG, erit BO, ad G _a_ parallela, etiam radii AB refractus. </s>
  <s xml:id="echoid-s1451" xml:space="preserve">Ergò <lb/>punctum A in recta OB protenſa apparebit. </s>
  <s xml:id="echoid-s1452" xml:space="preserve">Quoad hujuſmodi radi-<lb/>ationem nil ſuccurrit aliud; </s>
  <s xml:id="echoid-s1453" xml:space="preserve">itaque de propinquo radiantis puncti ſym-<lb/>ptomata contemplemur.</s>
  <s xml:id="echoid-s1454" xml:space="preserve"/>
</p>
<div xml:id="echoid-div42" type="float" level="2" n="2">
<note position="right" xlink:label="note-0049-02" xlink:href="note-0049-02a" xml:space="preserve">Fig. 32.</note>
</div>
<p>
  <s xml:id="echoid-s1455" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s1456" xml:space="preserve">Sit recta AB rectæ refringenti EF perpendicularis; </s>
  <s xml:id="echoid-s1457" xml:space="preserve">in qua <lb/>
<anchor type="note" xlink:label="note-0049-03a" xlink:href="note-0049-03"/>
ſit punctum radians A, ab EF haud ad ſenſum longè remotum; </s>
  <s xml:id="echoid-s1458" xml:space="preserve">ab hoc
<pb o="32" file="0050" n="50" rhead=""/>
autem procedentis cujuſvis radii (ceu AN) refractus N _a_ cum ipſa AB <lb/>
<anchor type="note" xlink:label="note-0050-01a" xlink:href="note-0050-01"/>
(protractus utique, vel retractus) conve@iat in K; </s>
  <s xml:id="echoid-s1459" xml:space="preserve">dico fore NK. <lb/></s>
  <s xml:id="echoid-s1460" xml:space="preserve">NA :</s>
  <s xml:id="echoid-s1461" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1462" xml:space="preserve">R. </s>
  <s xml:id="echoid-s1463" xml:space="preserve">(Neque non inverſè, ſi fuerit NK. </s>
  <s xml:id="echoid-s1464" xml:space="preserve">NA :</s>
  <s xml:id="echoid-s1465" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1466" xml:space="preserve">R; </s>
  <s xml:id="echoid-s1467" xml:space="preserve"><lb/>erit KN _a_ ipſius NA refractus.)</s>
  <s xml:id="echoid-s1468" xml:space="preserve"/>
</p>
<div xml:id="echoid-div43" type="float" level="2" n="3">
<note position="right" xlink:label="note-0049-03" xlink:href="note-0049-03a" xml:space="preserve">Fig. 33.</note>
<note position="left" xlink:label="note-0050-01" xlink:href="note-0050-01a" xml:space="preserve">_lect. 3. num. 9_.</note>
</div>
<p>
  <s xml:id="echoid-s1469" xml:space="preserve">Hoc è ſuperiùs oſtenſis immediatè conſectatur. </s>
  <s xml:id="echoid-s1470" xml:space="preserve">Et hinc etiam ſatis <lb/>apparet, quoniam (id quod bene notetur, ut paſſim in ſequentibus aſ-<lb/>ſumendum) angulus NAB, æ quatur angulo incidentiæ (quippe cum <lb/>is complementum ſit anguli ANB;) </s>
  <s xml:id="echoid-s1471" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1472" xml:space="preserve">angulus NKB (comple-<lb/>mentum videlicet anguli KNB) æquatur angulo refracto. </s>
  <s xml:id="echoid-s1473" xml:space="preserve">Cùm ita-<lb/>que ſit hinc ſinus anguli NAB (vel anguli deinceps NAK) ad ſi-<lb/>
<anchor type="note" xlink:label="note-0050-02a" xlink:href="note-0050-02"/>
num anguli NKA, ut I ad R; </s>
  <s xml:id="echoid-s1474" xml:space="preserve">etiam in triangulo NAK latus NK ad <lb/>latus NA ſeſe habebit ut I ad R. </s>
  <s xml:id="echoid-s1475" xml:space="preserve">Quod E. </s>
  <s xml:id="echoid-s1476" xml:space="preserve">D. </s>
  <s xml:id="echoid-s1477" xml:space="preserve">Quinetiam ſi latera <lb/>NK, NA ſe habeant ut I ad R; </s>
  <s xml:id="echoid-s1478" xml:space="preserve">etiam dictorum angulorum ſinus ità <lb/>ſe habebunt; </s>
  <s xml:id="echoid-s1479" xml:space="preserve">unde conſtabit ipſam KN _a_ ad AN pertinere.</s>
  <s xml:id="echoid-s1480" xml:space="preserve"/>
</p>
<div xml:id="echoid-div44" type="float" level="2" n="4">
<note position="left" xlink:label="note-0050-02" xlink:href="note-0050-02a" xml:space="preserve">Fig. 34, 35.</note>
</div>
<p>
  <s xml:id="echoid-s1481" xml:space="preserve">V. </s>
  <s xml:id="echoid-s1482" xml:space="preserve">Hinc particularis emergit expeditiſſimus modus hujuſmodi quot-<lb/>cunque refractos deſignandi. </s>
  <s xml:id="echoid-s1483" xml:space="preserve">Nempe per radians punctum A ducatur <lb/>AB refringenti EF perpendicularis; </s>
  <s xml:id="echoid-s1484" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1485" xml:space="preserve">fiat AB. </s>
  <s xml:id="echoid-s1486" xml:space="preserve">ZB :</s>
  <s xml:id="echoid-s1487" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s1488" xml:space="preserve">I; </s>
  <s xml:id="echoid-s1489" xml:space="preserve">tum <lb/>per Z ducatur recta GH ad EF parallela, Proponatur jam quilibet <lb/>incidens AN, cui conveniens deſignandus eſt refractus. </s>
  <s xml:id="echoid-s1490" xml:space="preserve">Eum ſic de-<lb/>deſignaveris. </s>
  <s xml:id="echoid-s1491" xml:space="preserve">Protrahatur NA (ſi opus) ut cum GH conveniat in <lb/>S; </s>
  <s xml:id="echoid-s1492" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1493" xml:space="preserve">centro N per S deſcribatur circulus ipſam AB ſecans in K (ſe-<lb/>cabit utique ſi refractus aliquis ad incidentem AN pertineat) erit con-<lb/>nexa KN, protractáque radio AN debitus refractus. </s>
  <s xml:id="echoid-s1494" xml:space="preserve">Etenim eſt <lb/>KN. </s>
  <s xml:id="echoid-s1495" xml:space="preserve">AN :</s>
  <s xml:id="echoid-s1496" xml:space="preserve">: SN. </s>
  <s xml:id="echoid-s1497" xml:space="preserve">AN :</s>
  <s xml:id="echoid-s1498" xml:space="preserve">: ZB. </s>
  <s xml:id="echoid-s1499" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s1500" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1501" xml:space="preserve">R :</s>
  <s xml:id="echoid-s1502" xml:space="preserve">: KN. </s>
  <s xml:id="echoid-s1503" xml:space="preserve">AN. </s>
  <s xml:id="echoid-s1504" xml:space="preserve">unde <lb/>liquet (è præcedente) propoſitum.</s>
  <s xml:id="echoid-s1505" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1506" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s1507" xml:space="preserve">Exhinc etiam hujuſmodi refractionis præcipua ſymptomata <lb/>perfacili colliguntur Negotio; </s>
  <s xml:id="echoid-s1508" xml:space="preserve">quæſeorſim acceptis, &amp; </s>
  <s xml:id="echoid-s1509" xml:space="preserve">quæ ſe-<lb/>cum mutuò collatis accidunt refractis; </s>
  <s xml:id="echoid-s1510" xml:space="preserve">hoc imprimis: </s>
  <s xml:id="echoid-s1511" xml:space="preserve">In primo caſu <lb/>(quum nempe refractio fit è rariori in denſius, ſeu quum I&amp;</s>
  <s xml:id="echoid-s1512" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s1513" xml:space="preserve">R) <lb/>concurſus refractorum cum recta AB (quam ſubinde radiationis hujus <lb/>axem appellare licebit) ſupra punctum Z exiſtit. </s>
  <s xml:id="echoid-s1514" xml:space="preserve">Nam connexâ NZ; <lb/></s>
  <s xml:id="echoid-s1515" xml:space="preserve">quoniam ang. </s>
  <s xml:id="echoid-s1516" xml:space="preserve">NZS recto BZS major eſt, erit NS (vel NK) &amp;</s>
  <s xml:id="echoid-s1517" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1518" xml:space="preserve"><lb/>NZ; </s>
  <s xml:id="echoid-s1519" xml:space="preserve">adeoque BK&amp;</s>
  <s xml:id="echoid-s1520" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s1521" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s1522" xml:space="preserve">Item, in ſecundo caſu (quum media <lb/>contrariè ſe habent) dictus concurſus infra punctum Z exiſtit. </s>
  <s xml:id="echoid-s1523" xml:space="preserve">Ete-<lb/>nim rurſus connexâ NZ; </s>
  <s xml:id="echoid-s1524" xml:space="preserve">eſt ang. </s>
  <s xml:id="echoid-s1525" xml:space="preserve">NSZ recto AZS (interno) ma-<lb/>jor, adeóque NZ &amp;</s>
  <s xml:id="echoid-s1526" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1527" xml:space="preserve">NS, vel NK; </s>
  <s xml:id="echoid-s1528" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1529" xml:space="preserve">ideò BZ &amp;</s>
  <s xml:id="echoid-s1530" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1531" xml:space="preserve">BK.</s>
  <s xml:id="echoid-s1532" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1533" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s1534" xml:space="preserve">Hinc liquet punctum Z eſſe limitem ultra vel citra quem (re-<lb/>ſpectivè) omnes refracti cum axe AB concurrunt. </s>
  <s xml:id="echoid-s1535" xml:space="preserve">Quinimò quòd <lb/>
<anchor type="note" xlink:label="note-0050-03a" xlink:href="note-0050-03"/>
<pb o="33" file="0051" n="51" rhead=""/>
ipſius perpendicularis AB (quaſi) refractus in ipſum punctum Z ter-<lb/>minatur. </s>
  <s xml:id="echoid-s1536" xml:space="preserve">Porrò:</s>
  <s xml:id="echoid-s1537" xml:space="preserve"/>
</p>
<div xml:id="echoid-div45" type="float" level="2" n="5">
<note position="left" xlink:label="note-0050-03" xlink:href="note-0050-03a" xml:space="preserve">Fig. 35.</note>
</div>
<p>
  <s xml:id="echoid-s1538" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s1539" xml:space="preserve">_Lemma_: </s>
  <s xml:id="echoid-s1540" xml:space="preserve">ſit AB ad EF normalis, &amp; </s>
  <s xml:id="echoid-s1541" xml:space="preserve">à duobus in AB <lb/>ſumptis utcunque punctis A, I (quorum A proprius ipſi B) ad duo <lb/>puncta quæ vis M, N in ipſa EF acceptis (quorum verò M ſit ipſi B <lb/>vicinius) connectantur rectæ AM, AN ; </s>
  <s xml:id="echoid-s1542" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1543" xml:space="preserve">IM, IN; </s>
  <s xml:id="echoid-s1544" xml:space="preserve">dico fore A N. <lb/></s>
  <s xml:id="echoid-s1545" xml:space="preserve">AM &amp;</s>
  <s xml:id="echoid-s1546" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1547" xml:space="preserve">IN. </s>
  <s xml:id="echoid-s1548" xml:space="preserve">IM.</s>
  <s xml:id="echoid-s1549" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1550" xml:space="preserve">Nam centro N per A deſcribatur circulus PA OR (rectas IM, <lb/>IN interſecans punctis O, R) &amp; </s>
  <s xml:id="echoid-s1551" xml:space="preserve">per R ducatur RT ad EF parallela, <lb/>ſecans IM in S. </s>
  <s xml:id="echoid-s1552" xml:space="preserve">Et ob angulum NRT obtuſum, patet rectam RT <lb/>extra circulum totam excidere; </s>
  <s xml:id="echoid-s1553" xml:space="preserve">unde SM &amp;</s>
  <s xml:id="echoid-s1554" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1555" xml:space="preserve">(OM &amp;</s>
  <s xml:id="echoid-s1556" xml:space="preserve">gt;) </s>
  <s xml:id="echoid-s1557" xml:space="preserve">AM. <lb/></s>
  <s xml:id="echoid-s1558" xml:space="preserve">adeóque AN. </s>
  <s xml:id="echoid-s1559" xml:space="preserve">AM &amp;</s>
  <s xml:id="echoid-s1560" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1561" xml:space="preserve">AN : </s>
  <s xml:id="echoid-s1562" xml:space="preserve">SM :</s>
  <s xml:id="echoid-s1563" xml:space="preserve">: RN. </s>
  <s xml:id="echoid-s1564" xml:space="preserve">SM :</s>
  <s xml:id="echoid-s1565" xml:space="preserve">: IN. </s>
  <s xml:id="echoid-s1566" xml:space="preserve">IM. </s>
  <s xml:id="echoid-s1567" xml:space="preserve">li-<lb/>quet igitur eſſe AN . </s>
  <s xml:id="echoid-s1568" xml:space="preserve">AM &amp;</s>
  <s xml:id="echoid-s1569" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1570" xml:space="preserve">IN. </s>
  <s xml:id="echoid-s1571" xml:space="preserve">IM : </s>
  <s xml:id="echoid-s1572" xml:space="preserve">Quod E. </s>
  <s xml:id="echoid-s1573" xml:space="preserve">D. </s>
  <s xml:id="echoid-s1574" xml:space="preserve">Hinc</s>
</p>
<p>
  <s xml:id="echoid-s1575" xml:space="preserve">Si duorum radiorum AM, AN (quorum hic obliquior) refracti <lb/>M _a_, N _a_ cum axe AB conveniant punctis I, K, erit in primo caſu IB <lb/>&amp;</s>
  <s xml:id="echoid-s1576" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s1577" xml:space="preserve">KB; </s>
  <s xml:id="echoid-s1578" xml:space="preserve">in ſecundo IB &amp;</s>
  <s xml:id="echoid-s1579" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1580" xml:space="preserve">KB. </s>
  <s xml:id="echoid-s1581" xml:space="preserve">Etenim connexâ IN; </s>
  <s xml:id="echoid-s1582" xml:space="preserve">eſt in primo <lb/>caſu, NK. </s>
  <s xml:id="echoid-s1583" xml:space="preserve">MI :</s>
  <s xml:id="echoid-s1584" xml:space="preserve">: NA. </s>
  <s xml:id="echoid-s1585" xml:space="preserve">MA &amp;</s>
  <s xml:id="echoid-s1586" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1587" xml:space="preserve">NI. </s>
  <s xml:id="echoid-s1588" xml:space="preserve">MI. </s>
  <s xml:id="echoid-s1589" xml:space="preserve">adeóque NK &amp;</s>
  <s xml:id="echoid-s1590" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1591" xml:space="preserve">NI. <lb/></s>
  <s xml:id="echoid-s1592" xml:space="preserve">unde BK &amp;</s>
  <s xml:id="echoid-s1593" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1594" xml:space="preserve">BI. </s>
  <s xml:id="echoid-s1595" xml:space="preserve">aſt in ſecundo, NK. </s>
  <s xml:id="echoid-s1596" xml:space="preserve">MI :</s>
  <s xml:id="echoid-s1597" xml:space="preserve">: NA. </s>
  <s xml:id="echoid-s1598" xml:space="preserve">MA &amp;</s>
  <s xml:id="echoid-s1599" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s1600" xml:space="preserve">NI. </s>
  <s xml:id="echoid-s1601" xml:space="preserve"><lb/>MI quare NK &amp;</s>
  <s xml:id="echoid-s1602" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s1603" xml:space="preserve">NI; </s>
  <s xml:id="echoid-s1604" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1605" xml:space="preserve">indè BK &amp;</s>
  <s xml:id="echoid-s1606" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s1607" xml:space="preserve">BI.</s>
  <s xml:id="echoid-s1608" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 37.</note>
<p>
  <s xml:id="echoid-s1609" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s1610" xml:space="preserve">_Coroll_. </s>
  <s xml:id="echoid-s1611" xml:space="preserve">Refractorum in primo caſu concurſus extra angulum <lb/>AB N verſantur; </s>
  <s xml:id="echoid-s1612" xml:space="preserve">in ſecundo, intra eundem. </s>
  <s xml:id="echoid-s1613" xml:space="preserve">Sed hæc eadem in decur-<lb/>ſu liquidius, ac multifariàm conſtabunt.</s>
  <s xml:id="echoid-s1614" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1615" xml:space="preserve">X. </s>
  <s xml:id="echoid-s1616" xml:space="preserve">Porrò, bina quoad hos caſus _Theoremata_ ſubjiciemus, uſus haud <lb/>contemnendi.</s>
  <s xml:id="echoid-s1617" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1618" xml:space="preserve">I. </s>
  <s xml:id="echoid-s1619" xml:space="preserve">Si ſiat (in primo caſu) YB. </s>
  <s xml:id="echoid-s1620" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s1621" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1622" xml:space="preserve">√ Iq - Rq. </s>
  <s xml:id="echoid-s1623" xml:space="preserve">ſit autem cu-<lb/>
<anchor type="note" xlink:label="note-0051-02a" xlink:href="note-0051-02"/>
juſvis incidentis AN reſractus KN _a_; </s>
  <s xml:id="echoid-s1624" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1625" xml:space="preserve">connectatur YN: </s>
  <s xml:id="echoid-s1626" xml:space="preserve">erit KB. <lb/></s>
  <s xml:id="echoid-s1627" xml:space="preserve">YN :</s>
  <s xml:id="echoid-s1628" xml:space="preserve">: √ Iq - Rq. </s>
  <s xml:id="echoid-s1629" xml:space="preserve">R.</s>
  <s xml:id="echoid-s1630" xml:space="preserve"/>
</p>
<div xml:id="echoid-div46" type="float" level="2" n="6">
<note position="right" xlink:label="note-0051-02" xlink:href="note-0051-02a" xml:space="preserve">Fig. 39.</note>
</div>
<p>
  <s xml:id="echoid-s1631" xml:space="preserve">Nam ob YBq. </s>
  <s xml:id="echoid-s1632" xml:space="preserve">ABq :</s>
  <s xml:id="echoid-s1633" xml:space="preserve">: (_a_) Iq. </s>
  <s xml:id="echoid-s1634" xml:space="preserve">Iq - Rq. </s>
  <s xml:id="echoid-s1635" xml:space="preserve">erit per converſi-<lb/>
<anchor type="note" xlink:label="note-0051-03a" xlink:href="note-0051-03"/>
onem rationis YBq. </s>
  <s xml:id="echoid-s1636" xml:space="preserve">YBq - ABq :</s>
  <s xml:id="echoid-s1637" xml:space="preserve">: Iq. </s>
  <s xml:id="echoid-s1638" xml:space="preserve">Rq :</s>
  <s xml:id="echoid-s1639" xml:space="preserve">: _(b)_ KNq. </s>
  <s xml:id="echoid-s1640" xml:space="preserve">ANq. <lb/></s>
  <s xml:id="echoid-s1641" xml:space="preserve">
<anchor type="note" xlink:label="note-0051-04a" xlink:href="note-0051-04"/>
&amp; </s>
  <s xml:id="echoid-s1642" xml:space="preserve">permutando YBq. </s>
  <s xml:id="echoid-s1643" xml:space="preserve">KNq :</s>
  <s xml:id="echoid-s1644" xml:space="preserve">: YBq - ABq. </s>
  <s xml:id="echoid-s1645" xml:space="preserve">ANq . </s>
  <s xml:id="echoid-s1646" xml:space="preserve">componen-<lb/>dóque YBq + KNq. </s>
  <s xml:id="echoid-s1647" xml:space="preserve">KNq :</s>
  <s xml:id="echoid-s1648" xml:space="preserve">: YBq + BNq. </s>
  <s xml:id="echoid-s1649" xml:space="preserve">ANq. </s>
  <s xml:id="echoid-s1650" xml:space="preserve">(nempe <lb/>YBq - ABq + ANq = YBq + BNq; </s>
  <s xml:id="echoid-s1651" xml:space="preserve">quoniam ANq -<lb/>ABq = BNq) Quarè runſus permutando eſt YBq + KNq. <lb/></s>
  <s xml:id="echoid-s1652" xml:space="preserve">YBq. </s>
  <s xml:id="echoid-s1653" xml:space="preserve">+ BNq :</s>
  <s xml:id="echoid-s1654" xml:space="preserve">: KNq. </s>
  <s xml:id="echoid-s1655" xml:space="preserve">ANq. </s>
  <s xml:id="echoid-s1656" xml:space="preserve">dividendoque KNq - BNq. </s>
  <s xml:id="echoid-s1657" xml:space="preserve"><lb/>YBq + BNq :</s>
  <s xml:id="echoid-s1658" xml:space="preserve">: KNq - ANq. </s>
  <s xml:id="echoid-s1659" xml:space="preserve">ANq; </s>
  <s xml:id="echoid-s1660" xml:space="preserve">hoc eſt KBq. </s>
  <s xml:id="echoid-s1661" xml:space="preserve">YNq <lb/>:</s>
  <s xml:id="echoid-s1662" xml:space="preserve">: Iq - Rq. </s>
  <s xml:id="echoid-s1663" xml:space="preserve">Rq: </s>
  <s xml:id="echoid-s1664" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s1665" xml:space="preserve">E. </s>
  <s xml:id="echoid-s1666" xml:space="preserve">D.</s>
  <s xml:id="echoid-s1667" xml:space="preserve"/>
</p>
<div xml:id="echoid-div47" type="float" level="2" n="7">
<note position="right" xlink:label="note-0051-03" xlink:href="note-0051-03a" xml:space="preserve">_(a) Hypoib._</note>
<note position="right" xlink:label="note-0051-04" xlink:href="note-0051-04a" xml:space="preserve">_(b) 4 hujus_</note>
</div>
<pb o="34" file="0052" n="52" rhead=""/>
<p>
  <s xml:id="echoid-s1668" xml:space="preserve">XI _corol_. </s>
  <s xml:id="echoid-s1669" xml:space="preserve">I. </s>
  <s xml:id="echoid-s1670" xml:space="preserve">Hinc ſi duo reſracti M _a_, N _a_ cum Axe AB conve-<lb/>niant in I, K; </s>
  <s xml:id="echoid-s1671" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1672" xml:space="preserve">à puncto Y ad incidentias ducantur rectæ YM, YN; <lb/></s>
  <s xml:id="echoid-s1673" xml:space="preserve">erit KB. </s>
  <s xml:id="echoid-s1674" xml:space="preserve">IB :</s>
  <s xml:id="echoid-s1675" xml:space="preserve">: YN. </s>
  <s xml:id="echoid-s1676" xml:space="preserve">YM. </s>
  <s xml:id="echoid-s1677" xml:space="preserve">Nam KBq. </s>
  <s xml:id="echoid-s1678" xml:space="preserve">YNq :</s>
  <s xml:id="echoid-s1679" xml:space="preserve">: Iq - Rq. </s>
  <s xml:id="echoid-s1680" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0052-01a" xlink:href="note-0052-01"/>
Rq :</s>
  <s xml:id="echoid-s1681" xml:space="preserve">: IBq. </s>
  <s xml:id="echoid-s1682" xml:space="preserve">YMq. </s>
  <s xml:id="echoid-s1683" xml:space="preserve">quare permutatim KBq. </s>
  <s xml:id="echoid-s1684" xml:space="preserve">IBq :</s>
  <s xml:id="echoid-s1685" xml:space="preserve">: YNq. <lb/></s>
  <s xml:id="echoid-s1686" xml:space="preserve">YM q.</s>
  <s xml:id="echoid-s1687" xml:space="preserve"/>
</p>
<div xml:id="echoid-div48" type="float" level="2" n="8">
<note position="left" xlink:label="note-0052-01" xlink:href="note-0052-01a" xml:space="preserve">Fig. 40.</note>
</div>
<p>
  <s xml:id="echoid-s1688" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s1689" xml:space="preserve">2. </s>
  <s xml:id="echoid-s1690" xml:space="preserve">Hinc etiam ſi refracti M I, NK conveniant in X; </s>
  <s xml:id="echoid-s1691" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1692" xml:space="preserve">de-<lb/>mìttatur XP ad AB parallela; </s>
  <s xml:id="echoid-s1693" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1694" xml:space="preserve">huic protractæ MY, NY occur-<lb/>rant in R, S; </s>
  <s xml:id="echoid-s1695" xml:space="preserve">erit NS = MR Nam X P. </s>
  <s xml:id="echoid-s1696" xml:space="preserve">SN :</s>
  <s xml:id="echoid-s1697" xml:space="preserve">: KB. </s>
  <s xml:id="echoid-s1698" xml:space="preserve">YN :</s>
  <s xml:id="echoid-s1699" xml:space="preserve">: <lb/>IB. </s>
  <s xml:id="echoid-s1700" xml:space="preserve">YM :</s>
  <s xml:id="echoid-s1701" xml:space="preserve">: XP. </s>
  <s xml:id="echoid-s1702" xml:space="preserve">RM. </s>
  <s xml:id="echoid-s1703" xml:space="preserve">cum itaque ſit X P. </s>
  <s xml:id="echoid-s1704" xml:space="preserve">SN :</s>
  <s xml:id="echoid-s1705" xml:space="preserve">: XP . </s>
  <s xml:id="echoid-s1706" xml:space="preserve">RM; <lb/></s>
  <s xml:id="echoid-s1707" xml:space="preserve">erìt SN = RM.</s>
  <s xml:id="echoid-s1708" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1709" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s1710" xml:space="preserve">2. </s>
  <s xml:id="echoid-s1711" xml:space="preserve">In ſecundo caſu ; </s>
  <s xml:id="echoid-s1712" xml:space="preserve">ſit cujuſvis incidentis AN refractus <lb/>KN _a_ &amp; </s>
  <s xml:id="echoid-s1713" xml:space="preserve">fiat YBq. </s>
  <s xml:id="echoid-s1714" xml:space="preserve">KBq :</s>
  <s xml:id="echoid-s1715" xml:space="preserve">: Rq. </s>
  <s xml:id="echoid-s1716" xml:space="preserve">Rq - Iq; </s>
  <s xml:id="echoid-s1717" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1718" xml:space="preserve">connectatur YN; <lb/></s>
  <s xml:id="echoid-s1719" xml:space="preserve">erit ABq. </s>
  <s xml:id="echoid-s1720" xml:space="preserve">YNq :</s>
  <s xml:id="echoid-s1721" xml:space="preserve">: Rq - Iq. </s>
  <s xml:id="echoid-s1722" xml:space="preserve">Iq.</s>
  <s xml:id="echoid-s1723" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1724" xml:space="preserve">Nam quia KBq = KNq - BNq = KNq - YNq + YBq; </s>
  <s xml:id="echoid-s1725" xml:space="preserve">erit <lb/>(hypotheſin perſequendo) YBq. </s>
  <s xml:id="echoid-s1726" xml:space="preserve">KNq + YBq - YNq :</s>
  <s xml:id="echoid-s1727" xml:space="preserve">: Rq. <lb/></s>
  <s xml:id="echoid-s1728" xml:space="preserve">Rq - Iq :</s>
  <s xml:id="echoid-s1729" xml:space="preserve">: ANq. </s>
  <s xml:id="echoid-s1730" xml:space="preserve">ANq - KNq. </s>
  <s xml:id="echoid-s1731" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1732" xml:space="preserve">per rationis converſionem <lb/>YBq. </s>
  <s xml:id="echoid-s1733" xml:space="preserve">YNq - KNq :</s>
  <s xml:id="echoid-s1734" xml:space="preserve">: ANq. </s>
  <s xml:id="echoid-s1735" xml:space="preserve">KNq. </s>
  <s xml:id="echoid-s1736" xml:space="preserve">(eſt autem YBq = <lb/>
<anchor type="note" xlink:label="note-0052-02a" xlink:href="note-0052-02"/>
YNq - BNq = YNq - ANq + ABq) ergò YNq - ANq <lb/>+ ABq. </s>
  <s xml:id="echoid-s1737" xml:space="preserve">YNq - KNq :</s>
  <s xml:id="echoid-s1738" xml:space="preserve">: ANq. </s>
  <s xml:id="echoid-s1739" xml:space="preserve">KNq (hoc eſt, anteceden-<lb/>tes &amp; </s>
  <s xml:id="echoid-s1740" xml:space="preserve">conſequentes adjungendo) :</s>
  <s xml:id="echoid-s1741" xml:space="preserve">: YNq + ABq. </s>
  <s xml:id="echoid-s1742" xml:space="preserve">YNq. </s>
  <s xml:id="echoid-s1743" xml:space="preserve">quare <lb/>dividendo ANq - KNq. </s>
  <s xml:id="echoid-s1744" xml:space="preserve">KNq :</s>
  <s xml:id="echoid-s1745" xml:space="preserve">: ABq. </s>
  <s xml:id="echoid-s1746" xml:space="preserve">YNq hoc eſt Rq -<lb/>Iq. </s>
  <s xml:id="echoid-s1747" xml:space="preserve">Iq :</s>
  <s xml:id="echoid-s1748" xml:space="preserve">: ABq. </s>
  <s xml:id="echoid-s1749" xml:space="preserve">YNq: </s>
  <s xml:id="echoid-s1750" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s1751" xml:space="preserve">E. </s>
  <s xml:id="echoid-s1752" xml:space="preserve">D.</s>
  <s xml:id="echoid-s1753" xml:space="preserve"/>
</p>
<div xml:id="echoid-div49" type="float" level="2" n="9">
<note position="left" xlink:label="note-0052-02" xlink:href="note-0052-02a" xml:space="preserve">Fig. 41.</note>
</div>
<p>
  <s xml:id="echoid-s1754" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s1755" xml:space="preserve">_Corol_. </s>
  <s xml:id="echoid-s1756" xml:space="preserve">1. </s>
  <s xml:id="echoid-s1757" xml:space="preserve">Hinc rurſus, ſi duo reſracti M _a_, N _a_ ſecent axem punctis <lb/>I, K; </s>
  <s xml:id="echoid-s1758" xml:space="preserve">ipſos autem ſe decuſſent puncto X; </s>
  <s xml:id="echoid-s1759" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1760" xml:space="preserve">ſiat YP. </s>
  <s xml:id="echoid-s1761" xml:space="preserve">XP :</s>
  <s xml:id="echoid-s1762" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s1763" xml:space="preserve">√ <lb/>Rq - Iq. </s>
  <s xml:id="echoid-s1764" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1765" xml:space="preserve">per Y ducantur MY R, NY S; </s>
  <s xml:id="echoid-s1766" xml:space="preserve">erit NS = MR.</s>
  <s xml:id="echoid-s1767" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1768" xml:space="preserve">Nam SB. </s>
  <s xml:id="echoid-s1769" xml:space="preserve">KB :</s>
  <s xml:id="echoid-s1770" xml:space="preserve">: YP. </s>
  <s xml:id="echoid-s1771" xml:space="preserve">XP :</s>
  <s xml:id="echoid-s1772" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s1773" xml:space="preserve">√ Rq - Iq. </s>
  <s xml:id="echoid-s1774" xml:space="preserve">quare AB. <lb/></s>
  <s xml:id="echoid-s1775" xml:space="preserve">SN :</s>
  <s xml:id="echoid-s1776" xml:space="preserve">: √ Rq - Iq. </s>
  <s xml:id="echoid-s1777" xml:space="preserve">I. </s>
  <s xml:id="echoid-s1778" xml:space="preserve">item RB. </s>
  <s xml:id="echoid-s1779" xml:space="preserve">IB :</s>
  <s xml:id="echoid-s1780" xml:space="preserve">: YP. </s>
  <s xml:id="echoid-s1781" xml:space="preserve">XP :</s>
  <s xml:id="echoid-s1782" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s1783" xml:space="preserve">√ Rq -<lb/>
<anchor type="note" xlink:label="note-0052-03a" xlink:href="note-0052-03"/>
Iq. </s>
  <s xml:id="echoid-s1784" xml:space="preserve">quare AB. </s>
  <s xml:id="echoid-s1785" xml:space="preserve">RM :</s>
  <s xml:id="echoid-s1786" xml:space="preserve">: √ Rq - Iq. </s>
  <s xml:id="echoid-s1787" xml:space="preserve">I. </s>
  <s xml:id="echoid-s1788" xml:space="preserve">ergò AB. </s>
  <s xml:id="echoid-s1789" xml:space="preserve">SN :</s>
  <s xml:id="echoid-s1790" xml:space="preserve">: AB. <lb/></s>
  <s xml:id="echoid-s1791" xml:space="preserve">RM. </s>
  <s xml:id="echoid-s1792" xml:space="preserve">quare SN = RM.</s>
  <s xml:id="echoid-s1793" xml:space="preserve"/>
</p>
<div xml:id="echoid-div50" type="float" level="2" n="10">
<note position="left" xlink:label="note-0052-03" xlink:href="note-0052-03a" xml:space="preserve">Fig. 42.</note>
</div>
<p>
  <s xml:id="echoid-s1794" xml:space="preserve">2. </s>
  <s xml:id="echoid-s1795" xml:space="preserve">Hinc SB. </s>
  <s xml:id="echoid-s1796" xml:space="preserve">RB :</s>
  <s xml:id="echoid-s1797" xml:space="preserve">: KB. </s>
  <s xml:id="echoid-s1798" xml:space="preserve">IB.</s>
  <s xml:id="echoid-s1799" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1800" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s1801" xml:space="preserve">Porrò, notandum eſt quò radii ab A manantes axi viciniores <lb/>f<unsure/>unt eò refractos ipſorum ſpiſſiùs incedere; </s>
  <s xml:id="echoid-s1802" xml:space="preserve">ſeu minora fore concur-<lb/>ſuum interſtitia; </s>
  <s xml:id="echoid-s1803" xml:space="preserve">ut nempe ſi in refringente EF ſumantur æqualia <lb/>intervalla MN, NO; </s>
  <s xml:id="echoid-s1804" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1805" xml:space="preserve">radiorum punctis M, N, O incidentium <lb/>refracti M _a_, N _a_, O_a_ cum axe concurrant punctis I, K, L ; </s>
  <s xml:id="echoid-s1806" xml:space="preserve">erit in-
<pb o="35" file="0053" n="53" rhead=""/>
tervallum IK minus ipſo KL ; </s>
  <s xml:id="echoid-s1807" xml:space="preserve">ſeu generalius efferendo, libere ſum-<lb/>ptis ipſis MN, NO ; </s>
  <s xml:id="echoid-s1808" xml:space="preserve">erit IK. </s>
  <s xml:id="echoid-s1809" xml:space="preserve">KL &amp;</s>
  <s xml:id="echoid-s1810" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s1811" xml:space="preserve">MN. </s>
  <s xml:id="echoid-s1812" xml:space="preserve">NO. </s>
  <s xml:id="echoid-s1813" xml:space="preserve">hoc verò non <lb/>aliter, opinor, elegantius quam ex adjunctis uno, vel altero Theore-<lb/>
<anchor type="note" xlink:label="note-0053-01a" xlink:href="note-0053-01"/>
mate conſtabit.</s>
  <s xml:id="echoid-s1814" xml:space="preserve"/>
</p>
<div xml:id="echoid-div51" type="float" level="2" n="11">
<note position="right" xlink:label="note-0053-01" xlink:href="note-0053-01a" xml:space="preserve">Fig. 43.</note>
</div>
<p>
  <s xml:id="echoid-s1815" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s1816" xml:space="preserve">In primo caſu; </s>
  <s xml:id="echoid-s1817" xml:space="preserve">ſit (ut antehac) ZB. </s>
  <s xml:id="echoid-s1818" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s1819" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1820" xml:space="preserve">R; </s>
  <s xml:id="echoid-s1821" xml:space="preserve">ſuper-<lb/>que diametro ZB conſtituatur ſemicirculus; </s>
  <s xml:id="echoid-s1822" xml:space="preserve">cui à puncto B adapte-<lb/>tur BD = BA ; </s>
  <s xml:id="echoid-s1823" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1824" xml:space="preserve">per puncta Z, D ducta recta refringenti occur-<lb/>rat in Y ; </s>
  <s xml:id="echoid-s1825" xml:space="preserve">tum ad ſemiaxes BZ, BY (centro nempe B, vertice Z) de-<lb/>ſcribatur Hyperbole HZG; </s>
  <s xml:id="echoid-s1826" xml:space="preserve">in hac autem ſumpto quolibet puncto S <lb/>ducantur SN ad AB, &amp; </s>
  <s xml:id="echoid-s1827" xml:space="preserve">SK ad EF parallelæ. </s>
  <s xml:id="echoid-s1828" xml:space="preserve">Denique ducantur <lb/>
<anchor type="note" xlink:label="note-0053-02a" xlink:href="note-0053-02"/>
A N, KN _a_ erit KM _a_ incidentis AN refractus.</s>
  <s xml:id="echoid-s1829" xml:space="preserve"/>
</p>
<div xml:id="echoid-div52" type="float" level="2" n="12">
<note position="right" xlink:label="note-0053-02" xlink:href="note-0053-02a" xml:space="preserve">Fig. 44.</note>
</div>
<p>
  <s xml:id="echoid-s1830" xml:space="preserve">Nam ex _Hyperbolæ_ natura eſt KBq - ZBq. </s>
  <s xml:id="echoid-s1831" xml:space="preserve">BNq :</s>
  <s xml:id="echoid-s1832" xml:space="preserve">: BZq. <lb/></s>
  <s xml:id="echoid-s1833" xml:space="preserve">BYq :</s>
  <s xml:id="echoid-s1834" xml:space="preserve">: ZDq. </s>
  <s xml:id="echoid-s1835" xml:space="preserve">BDq (hoc eſt) :</s>
  <s xml:id="echoid-s1836" xml:space="preserve">: ZBq - ABq. </s>
  <s xml:id="echoid-s1837" xml:space="preserve">ABq. </s>
  <s xml:id="echoid-s1838" xml:space="preserve">quare <lb/>componendo KBq - ZBq + BNq. </s>
  <s xml:id="echoid-s1839" xml:space="preserve">BNq :</s>
  <s xml:id="echoid-s1840" xml:space="preserve">: ZBq. </s>
  <s xml:id="echoid-s1841" xml:space="preserve">ABq hoc <lb/>eſt KNq - ZBq. </s>
  <s xml:id="echoid-s1842" xml:space="preserve">BNq :</s>
  <s xml:id="echoid-s1843" xml:space="preserve">: ZBq. </s>
  <s xml:id="echoid-s1844" xml:space="preserve">ABq. </s>
  <s xml:id="echoid-s1845" xml:space="preserve">permutandóque KNq <lb/>- ZBq. </s>
  <s xml:id="echoid-s1846" xml:space="preserve">ZBq :</s>
  <s xml:id="echoid-s1847" xml:space="preserve">: BNq. </s>
  <s xml:id="echoid-s1848" xml:space="preserve">ABq rurſuſque componendo KNq. </s>
  <s xml:id="echoid-s1849" xml:space="preserve"><lb/>ZBq :</s>
  <s xml:id="echoid-s1850" xml:space="preserve">: ANq . </s>
  <s xml:id="echoid-s1851" xml:space="preserve">ABq. </s>
  <s xml:id="echoid-s1852" xml:space="preserve">denuóque permutando KNq. </s>
  <s xml:id="echoid-s1853" xml:space="preserve">ANq :</s>
  <s xml:id="echoid-s1854" xml:space="preserve">: <lb/>ZBq. </s>
  <s xml:id="echoid-s1855" xml:space="preserve">ABq :</s>
  <s xml:id="echoid-s1856" xml:space="preserve">: Iq . </s>
  <s xml:id="echoid-s1857" xml:space="preserve">Rq. </s>
  <s xml:id="echoid-s1858" xml:space="preserve">quare KN. </s>
  <s xml:id="echoid-s1859" xml:space="preserve">AN :</s>
  <s xml:id="echoid-s1860" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s1861" xml:space="preserve">R. </s>
  <s xml:id="echoid-s1862" xml:space="preserve">ergo KN ip-<lb/>ſius AN reſractus erit: </s>
  <s xml:id="echoid-s1863" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s1864" xml:space="preserve">E. </s>
  <s xml:id="echoid-s1865" xml:space="preserve">D.</s>
  <s xml:id="echoid-s1866" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1867" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s1868" xml:space="preserve">Hinc refractorum cum axe concurſus (puta I, K, L) à ſe <lb/>diſtant intervallis ordinatim applicatarum ad _Hyperbolam_, puta recta-<lb/>rum, BZ, MR, NS, OT ; </s>
  <s xml:id="echoid-s1869" xml:space="preserve">vel ipſarum O, ZI, ZK, ZL. </s>
  <s xml:id="echoid-s1870" xml:space="preserve">Hæ ve-<lb/>rò (ceu paſſim notum, &amp; </s>
  <s xml:id="echoid-s1871" xml:space="preserve">à nobis aliquando generatim circa cunctas <lb/>hujuſmodi curvas oſtenſum eſt) in majori ratione creſcunt, quam ipſæ <lb/>BM, BN, BO ; </s>
  <s xml:id="echoid-s1872" xml:space="preserve">nempe ZL . </s>
  <s xml:id="echoid-s1873" xml:space="preserve">ZK &amp;</s>
  <s xml:id="echoid-s1874" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1875" xml:space="preserve">LT . </s>
  <s xml:id="echoid-s1876" xml:space="preserve">KS. </s>
  <s xml:id="echoid-s1877" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1878" xml:space="preserve">ZK. </s>
  <s xml:id="echoid-s1879" xml:space="preserve">ZI &amp;</s>
  <s xml:id="echoid-s1880" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s1881" xml:space="preserve">KS. <lb/></s>
  <s xml:id="echoid-s1882" xml:space="preserve">IR. </s>
  <s xml:id="echoid-s1883" xml:space="preserve">quare ſatìs liquet propoſitum. </s>
  <s xml:id="echoid-s1884" xml:space="preserve">Enimverò prope verticem Z <lb/>ordinatarum differentiæ perquam exiguæ ſunt; </s>
  <s xml:id="echoid-s1885" xml:space="preserve">ut bene multorum <lb/>perpendiculari AB adjacentium radiorum refracti velut è puncto <lb/>Z manare videantur ; </s>
  <s xml:id="echoid-s1886" xml:space="preserve">utcunque circa ipſum præcipuè conſtipantur.</s>
  <s xml:id="echoid-s1887" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1888" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s1889" xml:space="preserve">Haud abſimiliter, in ſecundo caſu, ſuper ipſa AB deſcriba-<lb/>tur ſemicirculus; </s>
  <s xml:id="echoid-s1890" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1891" xml:space="preserve">huic accommodetur BD = BZ; </s>
  <s xml:id="echoid-s1892" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1893" xml:space="preserve">connexa <lb/>protractáque AD refringenti occurrat ad Y ; </s>
  <s xml:id="echoid-s1894" xml:space="preserve">tum centro B ſemiaxi-<lb/>bus BZ, BY deſcribatur ellipſis HZG; </s>
  <s xml:id="echoid-s1895" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1896" xml:space="preserve">in hac accepto quocunque <lb/>
<anchor type="note" xlink:label="note-0053-03a" xlink:href="note-0053-03"/>
puncto S ducantur SN ad ZB, &amp; </s>
  <s xml:id="echoid-s1897" xml:space="preserve">SK ad EF parallela; </s>
  <s xml:id="echoid-s1898" xml:space="preserve">connectan-<lb/>tur denique rectæ AN , KN; </s>
  <s xml:id="echoid-s1899" xml:space="preserve">erit KN incidentis AN refractus. <lb/></s>
  <s xml:id="echoid-s1900" xml:space="preserve">Etenim ex ellipſis natura eſt KSq. </s>
  <s xml:id="echoid-s1901" xml:space="preserve">ZBq - SNq :</s>
  <s xml:id="echoid-s1902" xml:space="preserve">: BYq. </s>
  <s xml:id="echoid-s1903" xml:space="preserve">BZq <lb/>:</s>
  <s xml:id="echoid-s1904" xml:space="preserve">: BYq. </s>
  <s xml:id="echoid-s1905" xml:space="preserve">BDq :</s>
  <s xml:id="echoid-s1906" xml:space="preserve">: BAq. </s>
  <s xml:id="echoid-s1907" xml:space="preserve">ADq:</s>
  <s xml:id="echoid-s1908" xml:space="preserve">: BAq. </s>
  <s xml:id="echoid-s1909" xml:space="preserve">BAq - BZq. </s>
  <s xml:id="echoid-s1910" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1911" xml:space="preserve">per con-
<pb o="36" file="0054" n="54" rhead=""/>
verſam rationem KSq. </s>
  <s xml:id="echoid-s1912" xml:space="preserve">KSq - ZBq + SNq :</s>
  <s xml:id="echoid-s1913" xml:space="preserve">: BAq. </s>
  <s xml:id="echoid-s1914" xml:space="preserve">BZq. </s>
  <s xml:id="echoid-s1915" xml:space="preserve">hoc <lb/>eſt KSq. </s>
  <s xml:id="echoid-s1916" xml:space="preserve">KNq - ZBq :</s>
  <s xml:id="echoid-s1917" xml:space="preserve">: BAq. </s>
  <s xml:id="echoid-s1918" xml:space="preserve">ZBq. </s>
  <s xml:id="echoid-s1919" xml:space="preserve">quare permutando erit KSq. <lb/></s>
  <s xml:id="echoid-s1920" xml:space="preserve">BAq :</s>
  <s xml:id="echoid-s1921" xml:space="preserve">: KNq - ZBq. </s>
  <s xml:id="echoid-s1922" xml:space="preserve">ZBq. </s>
  <s xml:id="echoid-s1923" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1924" xml:space="preserve">compoſitè KSq + BAq. </s>
  <s xml:id="echoid-s1925" xml:space="preserve"><lb/>BAq :</s>
  <s xml:id="echoid-s1926" xml:space="preserve">: KNq ZBq. </s>
  <s xml:id="echoid-s1927" xml:space="preserve">hoc eſt ANq. </s>
  <s xml:id="echoid-s1928" xml:space="preserve">BAq :</s>
  <s xml:id="echoid-s1929" xml:space="preserve">: KNq. </s>
  <s xml:id="echoid-s1930" xml:space="preserve">ZBq. </s>
  <s xml:id="echoid-s1931" xml:space="preserve">qua-<lb/>re rurſus permutando eſt ANq. </s>
  <s xml:id="echoid-s1932" xml:space="preserve">KNq :</s>
  <s xml:id="echoid-s1933" xml:space="preserve">: BAq. </s>
  <s xml:id="echoid-s1934" xml:space="preserve">ZBq :</s>
  <s xml:id="echoid-s1935" xml:space="preserve">: Rq. </s>
  <s xml:id="echoid-s1936" xml:space="preserve">Jq. </s>
  <s xml:id="echoid-s1937" xml:space="preserve"><lb/>itaque AN. </s>
  <s xml:id="echoid-s1938" xml:space="preserve">KN :</s>
  <s xml:id="echoid-s1939" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s1940" xml:space="preserve">I. </s>
  <s xml:id="echoid-s1941" xml:space="preserve">unde patet KN ipſius AN refractum fore: </s>
  <s xml:id="echoid-s1942" xml:space="preserve"><lb/>Q. </s>
  <s xml:id="echoid-s1943" xml:space="preserve">E. </s>
  <s xml:id="echoid-s1944" xml:space="preserve">D.</s>
  <s xml:id="echoid-s1945" xml:space="preserve"/>
</p>
<div xml:id="echoid-div53" type="float" level="2" n="13">
<note position="right" xlink:label="note-0053-03" xlink:href="note-0053-03a" xml:space="preserve">Fig. 45.</note>
</div>
<p>
  <s xml:id="echoid-s1946" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s1947" xml:space="preserve">Exhinc, ut in priore caſu, patet quòd diſtantiæ (ZI, IK, <lb/>KL) concurſuum æquantur differentiis ipſarum ZB, RM, SN, TO <lb/>ordinatarum ad ellipſim. </s>
  <s xml:id="echoid-s1948" xml:space="preserve">Et quod ZI, ZK &amp;</s>
  <s xml:id="echoid-s1949" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s1950" xml:space="preserve">IR. </s>
  <s xml:id="echoid-s1951" xml:space="preserve">KS. </s>
  <s xml:id="echoid-s1952" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s1953" xml:space="preserve">c dif-<lb/>ferentiæ porrò dictæ circa verticem ellipſis Z admodum exiguæ ſunt, <lb/>adeóque propinquiorum axi radiorum refracti circa Z denſè congre-<lb/>gantur, &amp; </s>
  <s xml:id="echoid-s1954" xml:space="preserve">velut ab eo procedere videntur.</s>
  <s xml:id="echoid-s1955" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s1956" xml:space="preserve">XX Ex his tandem univerſis colligitur quòd puncti radiantis A <lb/>imago (reſpectu ſcilicet oculi centrum O habentis uſpiam in axe AB <lb/>conſtitutum) circa punctum Z conſiſtet. </s>
  <s xml:id="echoid-s1957" xml:space="preserve">Sit enim D δ diameter pu-<lb/>pillæ (illa nempe quæ in plano EAFO ) &amp; </s>
  <s xml:id="echoid-s1958" xml:space="preserve">per hujus extrema tranſe-<lb/>ant radiorum AM, A μ reſracti IMD, I μ δ ; </s>
  <s xml:id="echoid-s1959" xml:space="preserve">ſanè patet quòd nullius <lb/>obliquioris (ceu ipſius AN, vel A @) refractus oculum ingredi poterit ; <lb/></s>
  <s xml:id="echoid-s1960" xml:space="preserve">quin univerſi tales aliorſum digredientur, adeóque nec illi quicquam <lb/>adviſum attinebunt; </s>
  <s xml:id="echoid-s1961" xml:space="preserve">eique nil omnino conferent efficiendo quaquam, <lb/>nedum determinando. </s>
  <s xml:id="echoid-s1962" xml:space="preserve">Quinimò cùm viſus a ſolis afficiatur radiis in-<lb/>
<anchor type="note" xlink:label="note-0054-01a" xlink:href="note-0054-01"/>
tra ſpatium ZI axem interſecantibus, adeóque velut ab eo procedenti-<lb/>bus, intra ſpatium ZI neceſſariò verſabitur imago ; </s>
  <s xml:id="echoid-s1963" xml:space="preserve">quia verò ex his <lb/>qui circa Z concurrunt oculo rectiùs incidunt, ideóque præcipuâ vi <lb/>pollent; </s>
  <s xml:id="echoid-s1964" xml:space="preserve">cùm &amp; </s>
  <s xml:id="echoid-s1965" xml:space="preserve">ii (uti mox oſtendimus) ſpiſſiores ſint, &amp; </s>
  <s xml:id="echoid-s1966" xml:space="preserve">præ cæte-<lb/>ris confertim incedant ( id quod etiam nonnihil illorum vim adauget) <lb/>cùm etiam iidem faciliùs ab oculo rurſus in idem punctum recolligantur <lb/>(id quod poſthac aliquatenus oſtendemus; </s>
  <s xml:id="echoid-s1967" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s1968" xml:space="preserve">interim ex eo fit veriſimile, <lb/>quòd res per exiguum foramen ſpectatæ, radiis ſcilicet obliquioribus <lb/>excluſis, longè diſtinctiùs, apprehenduntur) quoniam, inquam, hæc <lb/>ita ſe habent, iis perpenſis omninò rationi conſentaneum eſt objectum <lb/>videri ceu radios projiciens à puncto Z, hoc eſt ejus imaginem inibi con-<lb/>ſiſtere Addo, quòd ob exilem pupillæ latitudinem, &amp; </s>
  <s xml:id="echoid-s1969" xml:space="preserve">propter ali-<lb/>quantam oculi diſtantiam à refringente; </s>
  <s xml:id="echoid-s1970" xml:space="preserve">totum ſpacium ZI perquam <lb/>anguſtum erit, &amp; </s>
  <s xml:id="echoid-s1971" xml:space="preserve">inſtar puncti merebitur exiſtimari: </s>
  <s xml:id="echoid-s1972" xml:space="preserve">quæ cuncta <lb/>propoſitum abunde videntur confirmare.</s>
  <s xml:id="echoid-s1973" xml:space="preserve"/>
</p>
<div xml:id="echoid-div54" type="float" level="2" n="14">
<note position="left" xlink:label="note-0054-01" xlink:href="note-0054-01a" xml:space="preserve">Fig 46.</note>
</div>
<pb o="37" file="0055" n="55" rhead=""/>
<p>
  <s xml:id="echoid-s1974" xml:space="preserve">XXI. </s>
  <s xml:id="echoid-s1975" xml:space="preserve">Accedit tamen ei penitiùs aſtruendo etiam experientia quâ <lb/>nempe compertum habetur; </s>
  <s xml:id="echoid-s1976" xml:space="preserve">quòd objectum (velut A) in aquâ ſitum, <lb/>oculo (O) perpendiculariter imminenti, ità diſtans videtur (puta ad <lb/>Z) ut ſit perpetuò AZ quadrans ipſius AB, id quod ratiociniis præ-<lb/>cedentibus exquiſitè congruit. </s>
  <s xml:id="echoid-s1977" xml:space="preserve">Etenim cum experientia docuerit in <lb/>refractionibus ex aqua factis in aerem, _Sinum anguli Incidentia ad_ <lb/>_Sinum anguli Refracti_ ſe habere circiter, ut 3 ad 4 ; </s>
  <s xml:id="echoid-s1978" xml:space="preserve">erit juxta conſtru-<lb/>ctionem præmiſſam ipſius ZB ad AB ratio ſubſeſquitertia; </s>
  <s xml:id="echoid-s1979" xml:space="preserve">ſeu hæc <lb/>ad illam ut 3 ad 4. </s>
  <s xml:id="echoid-s1980" xml:space="preserve">Quare nihil erat caaſæ cur hoc fretus experimento <lb/>
<anchor type="note" xlink:label="note-0055-01a" xlink:href="note-0055-01"/>
Præclariſſimus vir receptam de reſractione ſententiam impugnaret, &amp; </s>
  <s xml:id="echoid-s1981" xml:space="preserve"><lb/>exploderet; </s>
  <s xml:id="echoid-s1982" xml:space="preserve">at potiùs ut ei promptiùs accederet, aut firmius adhæreret, <lb/>expoſiti Phænomeni cauſam adeò perſpicuam, adeo neceſſariam ſugge-<lb/>renti. </s>
  <s xml:id="echoid-s1983" xml:space="preserve">quinimo perpendicularem ipſam (quod adeò valde vult, acriterque <lb/>contendit) è ſuperiore doctrinâ quadantenus infringi, decurtaríque (ter-<lb/>minatione ſaltem refringi, tametſi non ſitu) patebit ad illam attendenti.</s>
  <s xml:id="echoid-s1984" xml:space="preserve"/>
</p>
<div xml:id="echoid-div55" type="float" level="2" n="15">
<note position="right" xlink:label="note-0055-01" xlink:href="note-0055-01a" xml:space="preserve">_Iſ. Voſſ._</note>
</div>
<p>
  <s xml:id="echoid-s1985" xml:space="preserve">XXII Habetur itaque definitus imaginis ſitus, ob oculum in axe <lb/>collocatum. </s>
  <s xml:id="echoid-s1986" xml:space="preserve">Succedit ut idem præſtemus oculi gratiâ extra ipſum ubi-<lb/>cunque ſiti. </s>
  <s xml:id="echoid-s1987" xml:space="preserve">Sed priùs unum eſt quod opportunè moneamus, anteà <lb/>prætermiſſum; </s>
  <s xml:id="echoid-s1988" xml:space="preserve">eâdem ſcilicet operâ quoad radios convergentes ſimul <lb/>ac divergentes confici negotium. </s>
  <s xml:id="echoid-s1989" xml:space="preserve">Erunt enim ad punctum quodvis <lb/>(ceu A) tendentium radiorum refracti prorſus iidem eum illis, qui <lb/>divergentibus ab A convenient, modo cæteris manentibus invariatis <lb/>(refringente ſcilicet &amp; </s>
  <s xml:id="echoid-s1990" xml:space="preserve">puncto A deſignatum ſitum retinentibus) me-<lb/>dia concipiantur tranſpoſita. </s>
  <s xml:id="echoid-s1991" xml:space="preserve">Nimirum, exempli cauſâ, ſi NK ſit <lb/>refractus radii BN verſus A tendentis è raro in denſum; </s>
  <s xml:id="echoid-s1992" xml:space="preserve">erit itidem <lb/>NH ipſi KN in directum poſitus radii AN B, è raro in denſum <lb/>
<anchor type="note" xlink:label="note-0055-02a" xlink:href="note-0055-02"/>
(quæ nempe prioribus homogenea ſint) procedentis refractus. </s>
  <s xml:id="echoid-s1993" xml:space="preserve">Itaq; <lb/></s>
  <s xml:id="echoid-s1994" xml:space="preserve">quæ de radiis divergentibus oſtenſa ſunt, ea convergentibus, adhibito <lb/>juſto moderamine, pariter adaptari poſſunt; </s>
  <s xml:id="echoid-s1995" xml:space="preserve">in horum locum diver-<lb/>gentes reſpectivè congruos ſubrogando. </s>
  <s xml:id="echoid-s1996" xml:space="preserve">Quare nedum in hoc caſu, <lb/>ſed in omnibus qui ſequentur, de radiis ſolummodò divergentibus in-<lb/>ſtituemus ſermonem; </s>
  <s xml:id="echoid-s1997" xml:space="preserve">eò ſubintelligentes etiam convergentes ex hac <lb/>regula determinabiles referri. </s>
  <s xml:id="echoid-s1998" xml:space="preserve">Quæ ſanè compendio deſerviens obſer-<lb/>vatio generalibus iſtis ſupra delibatis meruit intertexi; </s>
  <s xml:id="echoid-s1999" xml:space="preserve">nec enim ad <lb/>hanc ſolam quæ præ manibus, aſt ad omnes æquè, quaſlibet ad ſuper-<lb/>ficies, radiorum inflectiones ſe extendit.</s>
  <s xml:id="echoid-s2000" xml:space="preserve"/>
</p>
<div xml:id="echoid-div56" type="float" level="2" n="16">
<note position="right" xlink:label="note-0055-02" xlink:href="note-0055-02a" xml:space="preserve">Fig. 47.</note>
</div>
<p>
  <s xml:id="echoid-s2001" xml:space="preserve">XXIII, Adſimilem &amp; </s>
  <s xml:id="echoid-s2002" xml:space="preserve">indè conſequentem (cum paralleli à punctó
<pb o="38" file="0056" n="56" rhead=""/>
provcniant infinitè diſſito) circa radios parallelos obſervatiunculam, <lb/>compendio ſervientem, etiam hîc tempeſtivum fuerit adjungere; </s>
  <s xml:id="echoid-s2003" xml:space="preserve">pa-<lb/>rallelorum nempe Convexis incidentium partibus radiorum inflexi, <lb/>quoad poſitionis directionem, iidem erunt cum inflexis ipſorum con-<lb/>cavis partibus incidentium; </s>
  <s xml:id="echoid-s2004" xml:space="preserve">modò tranſpoſita concipiantur media. <lb/></s>
  <s xml:id="echoid-s2005" xml:space="preserve">Quare parallelorum radiationes examinando nihil erit opus convexas <lb/>partes à concavis diſtinguere; </s>
  <s xml:id="echoid-s2006" xml:space="preserve">ſeu exinde caſus multiplicare. </s>
  <s xml:id="echoid-s2007" xml:space="preserve">Res è <lb/>poſthac dicendis clarior evadet. </s>
  <s xml:id="echoid-s2008" xml:space="preserve">His admonitis, de tabula jam manum ; </s>
  <s xml:id="echoid-s2009" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s2010" xml:space="preserve">quam propoſuimus inſtituendam proximè diſquiſitionem ſequenti <lb/>reſervamus.</s>
  <s xml:id="echoid-s2011" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div58" type="section" level="1" n="14">
<head xml:id="echoid-head17" xml:space="preserve"><emph style="sc">Lect.</emph>V.</head>
<p>
  <s xml:id="echoid-s2012" xml:space="preserve">I. </s>
  <s xml:id="echoid-s2013" xml:space="preserve">EO jam provecti ſumus, ut radiantis (à ſenſibiliter finita diſtan-<lb/>tia) puncti locum apparentem inveſtigemus, illum nempe qui <lb/>reſultat, è peracta ad planam ſuperficiem refractione; </s>
  <s xml:id="echoid-s2014" xml:space="preserve">nec non reſpe-<lb/>ctu viſus extra radiationis axem conſtituti. </s>
  <s xml:id="echoid-s2015" xml:space="preserve">Quorſum imprimis ſpectat, <lb/>ut rectam determinemus lineam, in qua locus ille verſatur; </s>
  <s xml:id="echoid-s2016" xml:space="preserve">tum ut <lb/>ſingulare deſignemus in illa recta punctum, circa quod exquiſitè con-<lb/>ſiſtit. </s>
  <s xml:id="echoid-s2017" xml:space="preserve">Utriuſque quæſiti gratiâ conficiendum, (imo penitius excutien-<lb/>endum) venit hujuſmodi _Problema_:</s>
  <s xml:id="echoid-s2018" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s2019" xml:space="preserve">_II_. </s>
  <s xml:id="echoid-s2020" xml:space="preserve">Dato puncto A, in poſitione datam rectam EF radiante, deſig-<lb/>nandus eſt incidens, qui per alterum tranſeat datum punctum.</s>
  <s xml:id="echoid-s2021" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2022" xml:space="preserve">III. </s>
  <s xml:id="echoid-s2023" xml:space="preserve">Si datum punctum alterum (puta jam K) in recta AB exiſtat, <lb/>
<anchor type="note" xlink:label="note-0056-01a" xlink:href="note-0056-01"/>
ad refringentem EF perpendiculari Problema planum erit, ac ità faci-<lb/>lè conficietur. </s>
  <s xml:id="echoid-s2024" xml:space="preserve">In primo caſu (quando ſcilicet I &amp;</s>
  <s xml:id="echoid-s2025" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2026" xml:space="preserve">R) fiat AB. </s>
  <s xml:id="echoid-s2027" xml:space="preserve">YB <lb/>:</s>
  <s xml:id="echoid-s2028" xml:space="preserve">: √ Iq - Rq. </s>
  <s xml:id="echoid-s2029" xml:space="preserve">I. </s>
  <s xml:id="echoid-s2030" xml:space="preserve">itemque fiat KB. </s>
  <s xml:id="echoid-s2031" xml:space="preserve">T :</s>
  <s xml:id="echoid-s2032" xml:space="preserve">: √ Iq - Rq. </s>
  <s xml:id="echoid-s2033" xml:space="preserve">R; </s>
  <s xml:id="echoid-s2034" xml:space="preserve">tum cen-<lb/>tro Y intervallo T deſcriptus circulus ipſam EF ſecet in N ; </s>
  <s xml:id="echoid-s2035" xml:space="preserve">conne-<lb/>ctantunrque AN, KN; </s>
  <s xml:id="echoid-s2036" xml:space="preserve">erit KN _a_ ipſius AN refractus.</s>
  <s xml:id="echoid-s2037" xml:space="preserve"/>
</p>
<div xml:id="echoid-div58" type="float" level="2" n="1">
<note position="left" xlink:label="note-0056-01" xlink:href="note-0056-01a" xml:space="preserve">Fig. 48, 49.</note>
</div>
<p>
  <s xml:id="echoid-s2038" xml:space="preserve">Itidem in ſecundo caſu (cùm I &amp;</s>
  <s xml:id="echoid-s2039" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s2040" xml:space="preserve">R) fiat KB. </s>
  <s xml:id="echoid-s2041" xml:space="preserve">YB :</s>
  <s xml:id="echoid-s2042" xml:space="preserve">: √ Rq. </s>
  <s xml:id="echoid-s2043" xml:space="preserve">-<lb/>Iq. </s>
  <s xml:id="echoid-s2044" xml:space="preserve">R. </s>
  <s xml:id="echoid-s2045" xml:space="preserve">AB. </s>
  <s xml:id="echoid-s2046" xml:space="preserve">T :</s>
  <s xml:id="echoid-s2047" xml:space="preserve">: √ Rq - Iq. </s>
  <s xml:id="echoid-s2048" xml:space="preserve">I. </s>
  <s xml:id="echoid-s2049" xml:space="preserve">centroque Y intervallo T deſcri-<lb/>batur circulus ipſi EF occurrens in N ; </s>
  <s xml:id="echoid-s2050" xml:space="preserve">critque rurſus KN _a_ ipſius
<pb o="39" file="0057" n="57" rhead=""/>
AN refractus. </s>
  <s xml:id="echoid-s2051" xml:space="preserve">Hæc autem è ſupra poſitis Theorematis abunde <lb/>conſtant.</s>
  <s xml:id="echoid-s2052" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">_10 &amp; 13 Lect. 4._</note>
<p>
  <s xml:id="echoid-s2053" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s2054" xml:space="preserve">Verum extra caſum hunc, &amp; </s>
  <s xml:id="echoid-s2055" xml:space="preserve">particulares alios nonnullos (quos <lb/>hìc certè nil attinet commemorare) generatim &amp; </s>
  <s xml:id="echoid-s2056" xml:space="preserve">illimitatè conceptum <lb/>Problema ſolidum eſt, plureſque duabus ſolutiones admittit; </s>
  <s xml:id="echoid-s2057" xml:space="preserve">id quod <lb/>facilè perſpicietur concipiendo punctum datum (puta X) in primo ca-<lb/>ſu extra angulum AB F jacere (vel intra eundem, in ſecundo) quo po-<lb/>ſito liquet è præcedentibus obtingere poſſe nonnunquam, ut duorum <lb/>
<anchor type="note" xlink:label="note-0057-02a" xlink:href="note-0057-02"/>
ad partes BF incidentium refractì concurrant ad X; </s>
  <s xml:id="echoid-s2058" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s2059" xml:space="preserve">alterius <lb/>unius ad partes BE incidentis reſractum etiam per idem X tranſire <lb/>quod cùm ſubinde, dico, contingere poſſit, indè certo conſequetur <lb/>_Problema_ ſolidum eſſe.</s>
  <s xml:id="echoid-s2060" xml:space="preserve"/>
</p>
<div xml:id="echoid-div59" type="float" level="2" n="2">
<note position="right" xlink:label="note-0057-02" xlink:href="note-0057-02a" xml:space="preserve">Fig. 50.</note>
</div>
<p>
  <s xml:id="echoid-s2061" xml:space="preserve">V. </s>
  <s xml:id="echoid-s2062" xml:space="preserve">Pro cujus ſolutione, primùm adnoto vix ullum _Problema_ dari <lb/>(præſertim è difficilioribus) quod non peculiarem lineam naturâ ſibi-<lb/>met appropriatam habeat, cujus deſcriptione quàm expeditè conſtrua-<lb/>tur; </s>
  <s xml:id="echoid-s2063" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2064" xml:space="preserve">quidem ità, ut ſimul indolem ſuam prodat ; </s>
  <s xml:id="echoid-s2065" xml:space="preserve">poſſibilitatem, <lb/>inquam, &amp; </s>
  <s xml:id="echoid-s2066" xml:space="preserve">impoſſibilitatem ſuam; </s>
  <s xml:id="echoid-s2067" xml:space="preserve">determinationes, &amp; </s>
  <s xml:id="echoid-s2068" xml:space="preserve">limitationes <lb/>neceſſarias; </s>
  <s xml:id="echoid-s2069" xml:space="preserve">caſuum &amp; </s>
  <s xml:id="echoid-s2070" xml:space="preserve">ſolutionum varietatem apertè monſtret, &amp; </s>
  <s xml:id="echoid-s2071" xml:space="preserve">ve-<lb/>lut ob oculos repreſentet. </s>
  <s xml:id="echoid-s2072" xml:space="preserve">In cujus qualis qualis obſervationis ſpeci-<lb/>men (alia quædam poſtmodùm exhibituri) imprimis lineam propone-<lb/>mus hujuſce Problematis executioni peculiariter accommodatam, hoc <lb/>modo promptè deſcribendam.</s>
  <s xml:id="echoid-s2073" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2074" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s2075" xml:space="preserve">Per radians punctum A ducatur ARS refringenti parallela; <lb/></s>
  <s xml:id="echoid-s2076" xml:space="preserve">
<anchor type="note" xlink:label="note-0057-03a" xlink:href="note-0057-03"/>
eidémque perpendicularis AB utrinque protendatur indefinitè. </s>
  <s xml:id="echoid-s2077" xml:space="preserve">Item <lb/>per datum alterum punctum X protendatur XR ad AB parallela: <lb/></s>
  <s xml:id="echoid-s2078" xml:space="preserve">Quinetiam facto AS. </s>
  <s xml:id="echoid-s2079" xml:space="preserve">AR :</s>
  <s xml:id="echoid-s2080" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s2081" xml:space="preserve">R; </s>
  <s xml:id="echoid-s2082" xml:space="preserve">per S extendatur SU ad AB pa-<lb/>rallela. </s>
  <s xml:id="echoid-s2083" xml:space="preserve">Quibus ſtantibus per A quotcunque tranſeant rectæ ſecantes <lb/>ipſam SU punctis H; </s>
  <s xml:id="echoid-s2084" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2085" xml:space="preserve">centro X, intervallis ipſas AH exæquanti-<lb/>bus, deſcribantur circuli ſecantes perpendicularem AB punctis K; </s>
  <s xml:id="echoid-s2086" xml:space="preserve"><lb/>demum per X, K ductæ lineæ cum ipſis HA conveniant in N. </s>
  <s xml:id="echoid-s2087" xml:space="preserve">Per <lb/>ejuſmodi quæ cunque puncta tranſibit propoſito noſtro deſerviens linea <lb/>(AN N) quam ſuſcepimus deſcribendam; </s>
  <s xml:id="echoid-s2088" xml:space="preserve">cujuſce nimirum cum re-<lb/>fringente FF interſectiones ipſiſſima ſunt incidentiæ puncta, quæ in-<lb/>dagamus (hæ autem ad unas rectæ AB partes (veluti ad F ) aliquando <lb/>duæ erunt; </s>
  <s xml:id="echoid-s2089" xml:space="preserve">ſubinde tantùm una, cù EF ſic effectam curvam tangit; </s>
  <s xml:id="echoid-s2090" xml:space="preserve"><lb/>quandoque nulla, cùm EF ultra tangentem dictam jacet; </s>
  <s xml:id="echoid-s2091" xml:space="preserve">ad alteras <lb/>ſaltem una erit; </s>
  <s xml:id="echoid-s2092" xml:space="preserve">quæ ſatis attendenti manifeſta futura ſubnoto tan-
<pb o="40" file="0058" n="58" rhead=""/>
tùm &amp; </s>
  <s xml:id="echoid-s2093" xml:space="preserve">levi pede prætereo; </s>
  <s xml:id="echoid-s2094" xml:space="preserve">quoniam aliunde mox apparitura) ſit, <lb/>inquam, ejuſmodi quælibet interſectio N; </s>
  <s xml:id="echoid-s2095" xml:space="preserve">dico fore XN, ipſius AN <lb/>
<anchor type="note" xlink:label="note-0058-01a" xlink:href="note-0058-01"/>
refractum. </s>
  <s xml:id="echoid-s2096" xml:space="preserve">Etenim eſt I. </s>
  <s xml:id="echoid-s2097" xml:space="preserve">R :</s>
  <s xml:id="echoid-s2098" xml:space="preserve">: AH. </s>
  <s xml:id="echoid-s2099" xml:space="preserve">AT. </s>
  <s xml:id="echoid-s2100" xml:space="preserve">hoc eſt ( quoniam AH, <lb/>KX ſunt ex conſtructione pares) I. </s>
  <s xml:id="echoid-s2101" xml:space="preserve">R :</s>
  <s xml:id="echoid-s2102" xml:space="preserve">: KX. </s>
  <s xml:id="echoid-s2103" xml:space="preserve">AT :</s>
  <s xml:id="echoid-s2104" xml:space="preserve">: NK. </s>
  <s xml:id="echoid-s2105" xml:space="preserve">NA. <lb/></s>
  <s xml:id="echoid-s2106" xml:space="preserve">unde maniſeſtum, è præmonſtratis, eſt propoſitum.</s>
  <s xml:id="echoid-s2107" xml:space="preserve"/>
</p>
<div xml:id="echoid-div60" type="float" level="2" n="3">
<note position="right" xlink:label="note-0057-03" xlink:href="note-0057-03a" xml:space="preserve">Fig. 51.</note>
<note position="left" xlink:label="note-0058-01" xlink:href="note-0058-01a" xml:space="preserve">Fig. 51.</note>
</div>
<p>
  <s xml:id="echoid-s2108" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s2109" xml:space="preserve">Veruntamen hujuſmodi conſtructiones _Geometrarum_ uſus <lb/>aut non libenter admittit, aut alias ſaltem exigit per lineas vulgo no-<lb/>tas, atque receptas; </s>
  <s xml:id="echoid-s2110" xml:space="preserve">itaque conſuetudini morem gerentes rem aliter <lb/>conficiemus; </s>
  <s xml:id="echoid-s2111" xml:space="preserve">huc utique faciens ſequens _Problema Lemmaticum_ præ-<lb/>mittentes: </s>
  <s xml:id="echoid-s2112" xml:space="preserve">Dato angulo recto XP F; </s>
  <s xml:id="echoid-s2113" xml:space="preserve">punctóque quovis Y; </s>
  <s xml:id="echoid-s2114" xml:space="preserve">per hoc <lb/>rectam duce e dati anguli cruribus occurentem, ſic ut ab iis intercep-<lb/>ta ſit a qualis datæ rectæ T. </s>
  <s xml:id="echoid-s2115" xml:space="preserve">‖ Expeditiſſimè quidem perſicitur hoc ope <lb/>_Concboidis_ alicujus polo Y deſcriptæ; </s>
  <s xml:id="echoid-s2116" xml:space="preserve">ſed enim quoniam &amp; </s>
  <s xml:id="echoid-s2117" xml:space="preserve">iſte modus <lb/>
<anchor type="note" xlink:label="note-0058-02a" xlink:href="note-0058-02"/>
hand ità Geometricus cenſetur; </s>
  <s xml:id="echoid-s2118" xml:space="preserve">adhuc iiſdem Geometris obſequentes <lb/>ità propoſitum exequemur. </s>
  <s xml:id="echoid-s2119" xml:space="preserve">Ducatur YB ad PF perpendicularis; </s>
  <s xml:id="echoid-s2120" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2121" xml:space="preserve"><lb/>_Aſymptotis_ PX, PB ducatur _Hyperbola_ per Y tranſiens (ſi quidem <lb/>punctum Y exiſtat extra angulum datum, aut iſtius oppoſita (ſc. </s>
  <s xml:id="echoid-s2122" xml:space="preserve">pun-<lb/>ctum Y ſit intra dictum angulum) tum centro Y intervallo datam T <lb/>æquante deſcriptus circulus _Hyperbolam_ inteerſecet in K; </s>
  <s xml:id="echoid-s2123" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2124" xml:space="preserve">à K de-<lb/>mittatur KL ad BP perpendicularis; </s>
  <s xml:id="echoid-s2125" xml:space="preserve">accipiatur autem BN = PL; <lb/></s>
  <s xml:id="echoid-s2126" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2127" xml:space="preserve">per NY trajiciatur recta NG. </s>
  <s xml:id="echoid-s2128" xml:space="preserve">dico factum; </s>
  <s xml:id="echoid-s2129" xml:space="preserve">vel eſſe NG parem <lb/>datæ T. </s>
  <s xml:id="echoid-s2130" xml:space="preserve">‖ Nam (ductâ YH ad PB parallelâ) ex _Hyperbolæ_ proprie-<lb/>tate eſt PL x LK = PB x BY. </s>
  <s xml:id="echoid-s2131" xml:space="preserve">adeóque cùm ſit ex conſtructione <lb/>BN = PL; </s>
  <s xml:id="echoid-s2132" xml:space="preserve">erit BN x LK :</s>
  <s xml:id="echoid-s2133" xml:space="preserve">: PB x BY. </s>
  <s xml:id="echoid-s2134" xml:space="preserve">adeóque BN. </s>
  <s xml:id="echoid-s2135" xml:space="preserve">BY :</s>
  <s xml:id="echoid-s2136" xml:space="preserve">: <lb/>PB. </s>
  <s xml:id="echoid-s2137" xml:space="preserve">LK. </s>
  <s xml:id="echoid-s2138" xml:space="preserve">eſt autem BN. </s>
  <s xml:id="echoid-s2139" xml:space="preserve">BY :</s>
  <s xml:id="echoid-s2140" xml:space="preserve">: DY. </s>
  <s xml:id="echoid-s2141" xml:space="preserve">DG. </s>
  <s xml:id="echoid-s2142" xml:space="preserve">ergo eſt PB. </s>
  <s xml:id="echoid-s2143" xml:space="preserve">LK :</s>
  <s xml:id="echoid-s2144" xml:space="preserve">: <lb/>DY. </s>
  <s xml:id="echoid-s2145" xml:space="preserve">DG. </s>
  <s xml:id="echoid-s2146" xml:space="preserve">quare cùm ſit PB = DY. </s>
  <s xml:id="echoid-s2147" xml:space="preserve">erit LK = DG. </s>
  <s xml:id="echoid-s2148" xml:space="preserve">adeóque <lb/>(pares LH, DP addendo, vel ſubtrahendo) eſt KH = GP. </s>
  <s xml:id="echoid-s2149" xml:space="preserve">quin-<lb/>etiam eſt YH = LB = PN (communem nempe PB, vel LN <lb/>addendo) Ergò patet fore YK(vel T) æqualem ipſi GN: </s>
  <s xml:id="echoid-s2150" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s2151" xml:space="preserve">E. </s>
  <s xml:id="echoid-s2152" xml:space="preserve">F.</s>
  <s xml:id="echoid-s2153" xml:space="preserve"/>
</p>
<div xml:id="echoid-div61" type="float" level="2" n="4">
<note position="left" xlink:label="note-0058-02" xlink:href="note-0058-02a" xml:space="preserve">Fig. 52.</note>
</div>
<p>
  <s xml:id="echoid-s2154" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s2155" xml:space="preserve">Notandum eſt autem in caſu, quando punctum Y intra datum <lb/>angulum XPF exiſtit, quòd circulus ille centro Y deſcriptus ſubinde <lb/>deſignatam hyperbolem binis punctis ſecabit (quod enim pluribus haud <lb/>quoquam ſecabit univerſim haud ità pridem circa tales ad eadem con-<lb/>vexas curvas oſtendimus) quo caſu patet duas obvenire propoſiti ſolu-<lb/>tiones, aliquando rurſus ille dictus circulus _Hyperbolen_ continget; </s>
  <s xml:id="echoid-s2156" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2157" xml:space="preserve"><lb/>tum una tantùm per Y duci poterit recta, datam T adæquans; </s>
  <s xml:id="echoid-s2158" xml:space="preserve">illa <lb/>ſcilicet omnium quæ per Y dato angulo interſeri poſſunt minima. <lb/></s>
  <s xml:id="echoid-s2159" xml:space="preserve">Quod ſi circulus Hyperbolæ non occurrat, _Problema_ prorſus {άΠο}ριστον
<pb o="41" file="0059" n="59" rhead=""/>
erit. </s>
  <s xml:id="echoid-s2160" xml:space="preserve">‖ Sin punctum Y extra datum angulum exiſtat, evidens eſt tan-<lb/>tùm uno modo problemati ſatisfactum iri; </s>
  <s xml:id="echoid-s2161" xml:space="preserve">quódque per alteram in-<lb/>erſectionem, &amp; </s>
  <s xml:id="echoid-s2162" xml:space="preserve">Y, ducta recta ad angulum pertinet dato verticalem. <lb/></s>
  <s xml:id="echoid-s2163" xml:space="preserve">hæc, inq̀uam, tantillùm attendenti manifeſtè conſtabunt; </s>
  <s xml:id="echoid-s2164" xml:space="preserve">nihil ut ſit <lb/>opus hic plura verba conſumere. </s>
  <s xml:id="echoid-s2165" xml:space="preserve">verùm ut in horum caſuum primo <lb/>conſtet (id quod pro ſequentibus ex uſu erit cognoſcere) quando <lb/>dictus circulus _byperbolem_ contingit; </s>
  <s xml:id="echoid-s2166" xml:space="preserve">ſeu quando tantùm una per <lb/>Y recta quantitatis ejuſdem interſeri poſſit, hoc adnectemus _Theo-_ <lb/>_rema._</s>
  <s xml:id="echoid-s2167" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2168" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s2169" xml:space="preserve">Si à puncto quovis Y intra rectum angulum XPF exiſtente <lb/>
<anchor type="note" xlink:label="note-0059-01a" xlink:href="note-0059-01"/>
demittantur ad ejuſdem anguli latera perpendiculares YB, YD; </s>
  <s xml:id="echoid-s2170" xml:space="preserve">ac <lb/>inter YB, YD proportione mediæ ſint rectæ BN, GD; </s>
  <s xml:id="echoid-s2171" xml:space="preserve">per puncta <lb/>N, Y, G tranſibit recta cunctarum minima, quæ per Y ductæ angu-<lb/>lum XPF ſubtendere poſſunt.</s>
  <s xml:id="echoid-s2172" xml:space="preserve"/>
</p>
<div xml:id="echoid-div62" type="float" level="2" n="5">
<note position="right" xlink:label="note-0059-01" xlink:href="note-0059-01a" xml:space="preserve">Fig. 54.</note>
</div>
<p>
  <s xml:id="echoid-s2173" xml:space="preserve">Quòd NYG ſit una recta patet, quoniam eſt YB. </s>
  <s xml:id="echoid-s2174" xml:space="preserve">BN :</s>
  <s xml:id="echoid-s2175" xml:space="preserve">: GD. <lb/></s>
  <s xml:id="echoid-s2176" xml:space="preserve">DY (ex conſtructione nimirum) porrò per Y tranſeat alia quæcunque <lb/>recta LYM; </s>
  <s xml:id="echoid-s2177" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2178" xml:space="preserve">NH ad GN, MH ad PF perpendiculares concur-<lb/>rant in H. </s>
  <s xml:id="echoid-s2179" xml:space="preserve">item HA ad NG parallela ducatur; </s>
  <s xml:id="echoid-s2180" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2181" xml:space="preserve">GS ad PF; </s>
  <s xml:id="echoid-s2182" xml:space="preserve"><lb/>denuóque connectatur GH. </s>
  <s xml:id="echoid-s2183" xml:space="preserve">Jam patet triangula GDY, YBN, <lb/>HMN, HMR ſimilia fore; </s>
  <s xml:id="echoid-s2184" xml:space="preserve">quódque proptereà eſt MN. </s>
  <s xml:id="echoid-s2185" xml:space="preserve">MR :</s>
  <s xml:id="echoid-s2186" xml:space="preserve">: <lb/>MN q. </s>
  <s xml:id="echoid-s2187" xml:space="preserve">MHq :</s>
  <s xml:id="echoid-s2188" xml:space="preserve">: DGq. </s>
  <s xml:id="echoid-s2189" xml:space="preserve">YDq. </s>
  <s xml:id="echoid-s2190" xml:space="preserve">item (ob BN, DG, YD {.</s>
  <s xml:id="echoid-s2191" xml:space="preserve">./.</s>
  <s xml:id="echoid-s2192" xml:space="preserve">.}) <lb/>eſt BN. </s>
  <s xml:id="echoid-s2193" xml:space="preserve">YD :</s>
  <s xml:id="echoid-s2194" xml:space="preserve">: DGq. </s>
  <s xml:id="echoid-s2195" xml:space="preserve">YDq. </s>
  <s xml:id="echoid-s2196" xml:space="preserve">hoc eſt YN. </s>
  <s xml:id="echoid-s2197" xml:space="preserve">YG (vel MN. </s>
  <s xml:id="echoid-s2198" xml:space="preserve">GS) <lb/>:</s>
  <s xml:id="echoid-s2199" xml:space="preserve">: DGq. </s>
  <s xml:id="echoid-s2200" xml:space="preserve">YDq. </s>
  <s xml:id="echoid-s2201" xml:space="preserve">ergò eſt MN. </s>
  <s xml:id="echoid-s2202" xml:space="preserve">MR :</s>
  <s xml:id="echoid-s2203" xml:space="preserve">: MN. </s>
  <s xml:id="echoid-s2204" xml:space="preserve">GS. </s>
  <s xml:id="echoid-s2205" xml:space="preserve">adeóque MR <lb/> = GS. </s>
  <s xml:id="echoid-s2206" xml:space="preserve">itaque major eſt GS ipsâ MT; </s>
  <s xml:id="echoid-s2207" xml:space="preserve">abeóque rectæ GH, LM <lb/>protractæ concurrent; </s>
  <s xml:id="echoid-s2208" xml:space="preserve">puta ad Z. </s>
  <s xml:id="echoid-s2209" xml:space="preserve">ergò LM. </s>
  <s xml:id="echoid-s2210" xml:space="preserve">GH :</s>
  <s xml:id="echoid-s2211" xml:space="preserve">: LZ. </s>
  <s xml:id="echoid-s2212" xml:space="preserve">GZ. </s>
  <s xml:id="echoid-s2213" xml:space="preserve"><lb/>verùm propter angulum LGH recto P majorem, eſt LZ &amp;</s>
  <s xml:id="echoid-s2214" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2215" xml:space="preserve">GZ. </s>
  <s xml:id="echoid-s2216" xml:space="preserve"><lb/>quare LM &amp;</s>
  <s xml:id="echoid-s2217" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2218" xml:space="preserve">GH. </s>
  <s xml:id="echoid-s2219" xml:space="preserve">aſt ob angulum rectum GNH eſt GH &amp;</s>
  <s xml:id="echoid-s2220" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2221" xml:space="preserve">GN. </s>
  <s xml:id="echoid-s2222" xml:space="preserve"><lb/>quare magìs eſt LM &amp;</s>
  <s xml:id="echoid-s2223" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2224" xml:space="preserve">GN. </s>
  <s xml:id="echoid-s2225" xml:space="preserve">eodémque modo quævis per Y ducta <lb/>major oſtendetur ipsâ GN : </s>
  <s xml:id="echoid-s2226" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s2227" xml:space="preserve">E. </s>
  <s xml:id="echoid-s2228" xml:space="preserve">D.</s>
  <s xml:id="echoid-s2229" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2230" xml:space="preserve">X. </s>
  <s xml:id="echoid-s2231" xml:space="preserve">Hinc etiam ſi GN ſit in ratione YB ad YN quarta proportio-<lb/>nalis; </s>
  <s xml:id="echoid-s2232" xml:space="preserve">erit GN minima. </s>
  <s xml:id="echoid-s2233" xml:space="preserve">nam indè conſequetur fore YB, BN, GD, <lb/>YD {.</s>
  <s xml:id="echoid-s2234" xml:space="preserve">./.</s>
  <s xml:id="echoid-s2235" xml:space="preserve">.}. </s>
  <s xml:id="echoid-s2236" xml:space="preserve">Etenim erit YNq. </s>
  <s xml:id="echoid-s2237" xml:space="preserve">YBq :</s>
  <s xml:id="echoid-s2238" xml:space="preserve">: GN. </s>
  <s xml:id="echoid-s2239" xml:space="preserve">YN. </s>
  <s xml:id="echoid-s2240" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2241" xml:space="preserve">dividendo <lb/>BNq. </s>
  <s xml:id="echoid-s2242" xml:space="preserve">YBq :</s>
  <s xml:id="echoid-s2243" xml:space="preserve">: GY. </s>
  <s xml:id="echoid-s2244" xml:space="preserve">YN :</s>
  <s xml:id="echoid-s2245" xml:space="preserve">: DY . </s>
  <s xml:id="echoid-s2246" xml:space="preserve">BN. </s>
  <s xml:id="echoid-s2247" xml:space="preserve">ac indè YBq x DY = <lb/>BNcub; </s>
  <s xml:id="echoid-s2248" xml:space="preserve">velDY = {BN cub/YBq}. </s>
  <s xml:id="echoid-s2249" xml:space="preserve">itáque DY eſt quarta proportionalis in ra-<lb/>tione YB ad BN.</s>
  <s xml:id="echoid-s2250" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2251" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s2252" xml:space="preserve">Subnotari poteſt autem, quòd minimæ GN propiores remo-
<pb o="42" file="0060" n="60" rhead=""/>
tioribus minores ſunt. </s>
  <s xml:id="echoid-s2253" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2254" xml:space="preserve">quod cuivis eâ majori binæ pares interſeri <lb/>poſſunt, ad ejus utramque partem ſingula. </s>
  <s xml:id="echoid-s2255" xml:space="preserve">nimirum hæc è ſuperiori <lb/>conſtructione luculentè patent; </s>
  <s xml:id="echoid-s2256" xml:space="preserve">pauxillùm expende Sodes; </s>
  <s xml:id="echoid-s2257" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2258" xml:space="preserve">per-<lb/>ſpicies; </s>
  <s xml:id="echoid-s2259" xml:space="preserve">operámque meam non deſiderabis.</s>
  <s xml:id="echoid-s2260" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2261" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s2262" xml:space="preserve">His præſtratis ad _Principale conſtruendum Problema_ reverti-<lb/>mur; </s>
  <s xml:id="echoid-s2263" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2264" xml:space="preserve">reliqua detexenda. </s>
  <s xml:id="echoid-s2265" xml:space="preserve">ſcilicet imprimis à dato puncto A pro-<lb/>
<anchor type="note" xlink:label="note-0060-01a" xlink:href="note-0060-01"/>
diens radius eſt deſignandus, cujus refractus per datum punctum X <lb/>tranſibit. </s>
  <s xml:id="echoid-s2266" xml:space="preserve">‖ hoc ità conficitur. </s>
  <s xml:id="echoid-s2267" xml:space="preserve">per A, X ducantur refringenti perpen-<lb/>diculares AB, XP. </s>
  <s xml:id="echoid-s2268" xml:space="preserve">tum in primo caſu fiat AB. </s>
  <s xml:id="echoid-s2269" xml:space="preserve">YB :</s>
  <s xml:id="echoid-s2270" xml:space="preserve">: √ Iq - Rq. <lb/></s>
  <s xml:id="echoid-s2271" xml:space="preserve">I. </s>
  <s xml:id="echoid-s2272" xml:space="preserve">neque non fiat XP. </s>
  <s xml:id="echoid-s2273" xml:space="preserve">T :</s>
  <s xml:id="echoid-s2274" xml:space="preserve">: √ Iq - Rq. </s>
  <s xml:id="echoid-s2275" xml:space="preserve">R . </s>
  <s xml:id="echoid-s2276" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2277" xml:space="preserve">per punctum Y <lb/>tranſadigatur recta NG ſubtendens angulum APF, &amp; </s>
  <s xml:id="echoid-s2278" xml:space="preserve">ipſam T ex-<lb/>æquans; </s>
  <s xml:id="echoid-s2279" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2280" xml:space="preserve">connectantur AN, XN. </s>
  <s xml:id="echoid-s2281" xml:space="preserve">dico factum; </s>
  <s xml:id="echoid-s2282" xml:space="preserve">ſeu rectam XV <lb/>incidentis AN refractum eſſe. </s>
  <s xml:id="echoid-s2283" xml:space="preserve">‖ Etenim eſt X P. </s>
  <s xml:id="echoid-s2284" xml:space="preserve">KB :</s>
  <s xml:id="echoid-s2285" xml:space="preserve">: NP. </s>
  <s xml:id="echoid-s2286" xml:space="preserve">NB :</s>
  <s xml:id="echoid-s2287" xml:space="preserve">: <lb/>NG. </s>
  <s xml:id="echoid-s2288" xml:space="preserve">NY. </s>
  <s xml:id="echoid-s2289" xml:space="preserve">permutandóque XP. </s>
  <s xml:id="echoid-s2290" xml:space="preserve">NG :</s>
  <s xml:id="echoid-s2291" xml:space="preserve">: KB. </s>
  <s xml:id="echoid-s2292" xml:space="preserve">NY. </s>
  <s xml:id="echoid-s2293" xml:space="preserve">hoc eſt X P. </s>
  <s xml:id="echoid-s2294" xml:space="preserve"><lb/>NG (vel √ Iq - Rq. </s>
  <s xml:id="echoid-s2295" xml:space="preserve">R) :</s>
  <s xml:id="echoid-s2296" xml:space="preserve">: KB. </s>
  <s xml:id="echoid-s2297" xml:space="preserve">YN. </s>
  <s xml:id="echoid-s2298" xml:space="preserve">itaque per theorema præ-<lb/>miſſum liquet KN ipſius AN refractum eſſe: </s>
  <s xml:id="echoid-s2299" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s2300" xml:space="preserve">EF.</s>
  <s xml:id="echoid-s2301" xml:space="preserve"/>
</p>
<div xml:id="echoid-div63" type="float" level="2" n="6">
<note position="left" xlink:label="note-0060-01" xlink:href="note-0060-01a" xml:space="preserve">Fig. 55, 56.</note>
</div>
<p>
  <s xml:id="echoid-s2302" xml:space="preserve">Haud abſimiliter in ſecundo caſu; </s>
  <s xml:id="echoid-s2303" xml:space="preserve">fiat X P. </s>
  <s xml:id="echoid-s2304" xml:space="preserve">YP :</s>
  <s xml:id="echoid-s2305" xml:space="preserve">: √ Rq - Iq. </s>
  <s xml:id="echoid-s2306" xml:space="preserve">R; <lb/></s>
  <s xml:id="echoid-s2307" xml:space="preserve">itémque AB. </s>
  <s xml:id="echoid-s2308" xml:space="preserve">T :</s>
  <s xml:id="echoid-s2309" xml:space="preserve">: √ Rq - Iq. </s>
  <s xml:id="echoid-s2310" xml:space="preserve">I; </s>
  <s xml:id="echoid-s2311" xml:space="preserve">angulóque ABF per Y tranſi-<lb/>ens, ipſámque T adæquans inſeratur recta N G; </s>
  <s xml:id="echoid-s2312" xml:space="preserve">connectantúrque <lb/>AN, XN; </s>
  <s xml:id="echoid-s2313" xml:space="preserve">factum erit. </s>
  <s xml:id="echoid-s2314" xml:space="preserve">‖ Nam ipſam XP protractam ſecet AN in <lb/>S. </s>
  <s xml:id="echoid-s2315" xml:space="preserve">èſtque SP. </s>
  <s xml:id="echoid-s2316" xml:space="preserve">YN :</s>
  <s xml:id="echoid-s2317" xml:space="preserve">: AB. </s>
  <s xml:id="echoid-s2318" xml:space="preserve">GN :</s>
  <s xml:id="echoid-s2319" xml:space="preserve">: AB. </s>
  <s xml:id="echoid-s2320" xml:space="preserve">T :</s>
  <s xml:id="echoid-s2321" xml:space="preserve">: √ Rq - Iq. </s>
  <s xml:id="echoid-s2322" xml:space="preserve">I . </s>
  <s xml:id="echoid-s2323" xml:space="preserve">unde <lb/>conſequitur è præmonſtratis fore XN ipſius SN refractum Q. </s>
  <s xml:id="echoid-s2324" xml:space="preserve">E . </s>
  <s xml:id="echoid-s2325" xml:space="preserve">F .</s>
  <s xml:id="echoid-s2326" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2327" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s2328" xml:space="preserve">Exhinc, &amp; </s>
  <s xml:id="echoid-s2329" xml:space="preserve">præmiſſa reſpiciendo ſatìs diluceſcit non ultra <lb/>duos ad unas perpendicularis AB partes incidentium refractos in uno <lb/>puncto convenire. </s>
  <s xml:id="echoid-s2330" xml:space="preserve">nam (ut ſupra declaratum) per puntum Y (quod <lb/>univerſis hujuſmodi conſtructionibus commune, vel invariatum per-<lb/>perſiſtit; </s>
  <s xml:id="echoid-s2331" xml:space="preserve">in primo caſu quoad omnes ab A incidentes; </s>
  <s xml:id="echoid-s2332" xml:space="preserve">in ſecundo <lb/>qnoad omnes per X tranſeuntes refractos) plures duabus ſibi pares <lb/>duabus ſibi pares rectæ angulo recto XP F, vel ABF interſeri ne-<lb/>queunt; </s>
  <s xml:id="echoid-s2333" xml:space="preserve">adeóque nec plures refracti per ipſum X tranſibunt.</s>
  <s xml:id="echoid-s2334" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2335" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s2336" xml:space="preserve">Porrò, cùm è dictis definita habeatur recta, in qua puncti <lb/>A Imago verſatur; </s>
  <s xml:id="echoid-s2337" xml:space="preserve">iſte nimirum refractus qui per oculi centrum <lb/>tranſit, modo jam expoſito ducendus; </s>
  <s xml:id="echoid-s2338" xml:space="preserve">ipſum jam punctum determi-<lb/>nandum venit, ad quod illa præcisè conſiſtit; </s>
  <s xml:id="echoid-s2339" xml:space="preserve">id quod etiam è præce-<lb/>dentibus haud difficulter eliciemus.</s>
  <s xml:id="echoid-s2340" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2341" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s2342" xml:space="preserve">Sumatur, in caſu primo, punctum Y conditione præditum
<pb o="43" file="0061" n="61" rhead=""/>
jam aliquoties inſinuatâ; </s>
  <s xml:id="echoid-s2343" xml:space="preserve">ſcilicet ut ſit AB. </s>
  <s xml:id="echoid-s2344" xml:space="preserve">YB :</s>
  <s xml:id="echoid-s2345" xml:space="preserve">: √ Iq -Rq. </s>
  <s xml:id="echoid-s2346" xml:space="preserve">I; <lb/></s>
  <s xml:id="echoid-s2347" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2348" xml:space="preserve">deſignetur quilibet refractus KN; </s>
  <s xml:id="echoid-s2349" xml:space="preserve">tum continuetur ratio YB ad <lb/>BN; </s>
  <s xml:id="echoid-s2350" xml:space="preserve">ut ſit ad has proportione quarta BP; </s>
  <s xml:id="echoid-s2351" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2352" xml:space="preserve">per punctum P du-<lb/>
<anchor type="note" xlink:label="note-0061-01a" xlink:href="note-0061-01"/>
catur recta PZ ad AB parallela; </s>
  <s xml:id="echoid-s2353" xml:space="preserve">refracto KN occurrens in Z; </s>
  <s xml:id="echoid-s2354" xml:space="preserve">dico <lb/>nullum alium refractum per Z traſire. </s>
  <s xml:id="echoid-s2355" xml:space="preserve">Nam ſi ſieri poteſt tranſeat <lb/>alius ZR; </s>
  <s xml:id="echoid-s2356" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2357" xml:space="preserve">per Y traducantur rectæ NYG, RYS ; </s>
  <s xml:id="echoid-s2358" xml:space="preserve">è præmon-<lb/>ſtratis apparet quòd ſit RS<anchor type="note" xlink:href="" symbol="*"/> = NG. </s>
  <s xml:id="echoid-s2359" xml:space="preserve">item è prædictis manifeſtum eſt
<anchor type="note" xlink:label="note-0061-02a" xlink:href="note-0061-02"/>
quò RS<anchor type="note" xlink:href="" symbol="*"/> &amp;</s>
  <s xml:id="echoid-s2360" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2361" xml:space="preserve">N G. </s>
  <s xml:id="echoid-s2362" xml:space="preserve">quæ repugnant.</s>
  <s xml:id="echoid-s2363" xml:space="preserve"/>
</p>
<div xml:id="echoid-div64" type="float" level="2" n="7">
<note position="right" xlink:label="note-0061-01" xlink:href="note-0061-01a" xml:space="preserve">Fig. 57, 58.</note>
<note symbol="*" position="right" xlink:label="note-0061-02" xlink:href="note-0061-02a" xml:space="preserve">12. Lect. 4.</note>
</div>
<note symbol="*" position="right" xml:space="preserve">9 hujus Lect.</note>
<p>
  <s xml:id="echoid-s2364" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s2365" xml:space="preserve">Non diſpari ratione, quoad caſum ſecundum, deſignetur quilibet <lb/>refractus KN; </s>
  <s xml:id="echoid-s2366" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2367" xml:space="preserve">fiat KB . </s>
  <s xml:id="echoid-s2368" xml:space="preserve">GB :</s>
  <s xml:id="echoid-s2369" xml:space="preserve">: √ Rq - Iq. </s>
  <s xml:id="echoid-s2370" xml:space="preserve">R; </s>
  <s xml:id="echoid-s2371" xml:space="preserve">tum adnexâ <lb/>GN, ad ipſas NG, GB ſumatur tertia proportionalis V; </s>
  <s xml:id="echoid-s2372" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2373" xml:space="preserve">ſiat <lb/>NG. </s>
  <s xml:id="echoid-s2374" xml:space="preserve">V :</s>
  <s xml:id="echoid-s2375" xml:space="preserve">: BN. </s>
  <s xml:id="echoid-s2376" xml:space="preserve">NP; </s>
  <s xml:id="echoid-s2377" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2378" xml:space="preserve">per punctum P ducatur PY ad BA pa-<lb/>rallela refractum NK decuſſans in Z; </s>
  <s xml:id="echoid-s2379" xml:space="preserve">dico nullum alium refractum <lb/>per ipſum Z meare. </s>
  <s xml:id="echoid-s2380" xml:space="preserve">Nam, ſi neges, tranſeat alius ZR; </s>
  <s xml:id="echoid-s2381" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2382" xml:space="preserve">per <lb/>Y trajiciatur RY S; </s>
  <s xml:id="echoid-s2383" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2384" xml:space="preserve">quoniam ZP . </s>
  <s xml:id="echoid-s2385" xml:space="preserve">YP :</s>
  <s xml:id="echoid-s2386" xml:space="preserve">: KB. </s>
  <s xml:id="echoid-s2387" xml:space="preserve">GB :</s>
  <s xml:id="echoid-s2388" xml:space="preserve">: √ Rq <lb/>- Iq. </s>
  <s xml:id="echoid-s2389" xml:space="preserve">R. </s>
  <s xml:id="echoid-s2390" xml:space="preserve">ex * antedictis apparet fore RS = NG. </s>
  <s xml:id="echoid-s2391" xml:space="preserve">quinetiam ob <lb/>
<anchor type="note" xlink:label="note-0061-04a" xlink:href="note-0061-04"/>
NGq . </s>
  <s xml:id="echoid-s2392" xml:space="preserve">GBq :</s>
  <s xml:id="echoid-s2393" xml:space="preserve">: NG. </s>
  <s xml:id="echoid-s2394" xml:space="preserve">V :</s>
  <s xml:id="echoid-s2395" xml:space="preserve">: BN . </s>
  <s xml:id="echoid-s2396" xml:space="preserve">NP . </s>
  <s xml:id="echoid-s2397" xml:space="preserve">erit dividendo NBq. <lb/></s>
  <s xml:id="echoid-s2398" xml:space="preserve">GBq :</s>
  <s xml:id="echoid-s2399" xml:space="preserve">: BP . </s>
  <s xml:id="echoid-s2400" xml:space="preserve">NP . </s>
  <s xml:id="echoid-s2401" xml:space="preserve">hoc eſt NPq. </s>
  <s xml:id="echoid-s2402" xml:space="preserve">PYq :</s>
  <s xml:id="echoid-s2403" xml:space="preserve">: BP . </s>
  <s xml:id="echoid-s2404" xml:space="preserve">NP; </s>
  <s xml:id="echoid-s2405" xml:space="preserve">inde facile <lb/>deducitur eſſe BP quartam proportionalem in ratione YP ad PN ; </s>
  <s xml:id="echoid-s2406" xml:space="preserve"><lb/>conſequentérque fore RS minimâ NG majorem . </s>
  <s xml:id="echoid-s2407" xml:space="preserve">quod adver-<lb/>ſatur oſtenſis . </s>
  <s xml:id="echoid-s2408" xml:space="preserve">itaque potiùs per Z nullus alius tranſit reſractus: </s>
  <s xml:id="echoid-s2409" xml:space="preserve"><lb/>Q. </s>
  <s xml:id="echoid-s2410" xml:space="preserve">E. </s>
  <s xml:id="echoid-s2411" xml:space="preserve">D.</s>
  <s xml:id="echoid-s2412" xml:space="preserve"/>
</p>
<div xml:id="echoid-div65" type="float" level="2" n="8">
<note position="right" xlink:label="note-0061-04" xlink:href="note-0061-04a" xml:space="preserve">* 14. Lect.4.</note>
</div>
<p>
  <s xml:id="echoid-s2413" xml:space="preserve">XV I. </s>
  <s xml:id="echoid-s2414" xml:space="preserve">Prætereà, ſi refractum NKZ interſecet alius quilibet M I, <lb/>ad rectiorem pertinens incidentem (hoc eſt ut incidentiæ punctum M <lb/>inter B, &amp; </s>
  <s xml:id="echoid-s2415" xml:space="preserve">N jaceat) interſectio X ſolitario puncto Z citerior erit <lb/>(ſeu perpendiculari KB propinquior) . </s>
  <s xml:id="echoid-s2416" xml:space="preserve">Nam ab X demittatur per-<lb/>
<anchor type="note" xlink:label="note-0061-05a" xlink:href="note-0061-05"/>
pendicularis XQ. </s>
  <s xml:id="echoid-s2417" xml:space="preserve">ipſam NG ſecans in γ ; </s>
  <s xml:id="echoid-s2418" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2419" xml:space="preserve">(in primo caſu) per <lb/>M, Y traducatur recta MY H. </s>
  <s xml:id="echoid-s2420" xml:space="preserve">ergò MH = Nγ. </s>
  <s xml:id="echoid-s2421" xml:space="preserve">quare minima earum <lb/>quæ per Y angulo XQF interſeri poſſuntinter puncta M, N cadet <lb/>(utì nuper admonitum, &amp; </s>
  <s xml:id="echoid-s2422" xml:space="preserve">adſtructum). </s>
  <s xml:id="echoid-s2423" xml:space="preserve">puta ad φ. </s>
  <s xml:id="echoid-s2424" xml:space="preserve">ergò quum ſit <lb/>BP quarta proportionalis in ratione YB ad BN ; </s>
  <s xml:id="echoid-s2425" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2426" xml:space="preserve">BQ quarta <lb/>proportionalis in ratione YB ad B φ, erit PB &amp;</s>
  <s xml:id="echoid-s2427" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2428" xml:space="preserve">QB; </s>
  <s xml:id="echoid-s2429" xml:space="preserve">adeóque <lb/>recta XQ rectis ZP, KB interjacet : </s>
  <s xml:id="echoid-s2430" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s2431" xml:space="preserve">E . </s>
  <s xml:id="echoid-s2432" xml:space="preserve">D.</s>
  <s xml:id="echoid-s2433" xml:space="preserve"/>
</p>
<div xml:id="echoid-div66" type="float" level="2" n="9">
<note position="right" xlink:label="note-0061-05" xlink:href="note-0061-05a" xml:space="preserve">Fig. 59, 60.</note>
</div>
<p>
  <s xml:id="echoid-s2434" xml:space="preserve">In ſecundo caſu, per γ trajiciatur recta M γ H. </s>
  <s xml:id="echoid-s2435" xml:space="preserve">ergò cùm ſit <lb/>Q X. </s>
  <s xml:id="echoid-s2436" xml:space="preserve">Q γ :</s>
  <s xml:id="echoid-s2437" xml:space="preserve">: PZ. </s>
  <s xml:id="echoid-s2438" xml:space="preserve">PY :</s>
  <s xml:id="echoid-s2439" xml:space="preserve">: √ Rq - Iq. </s>
  <s xml:id="echoid-s2440" xml:space="preserve">R. </s>
  <s xml:id="echoid-s2441" xml:space="preserve">erit HM = GN. </s>
  <s xml:id="echoid-s2442" xml:space="preserve">ergò <lb/>minima per γ ducibilium angulo AB F intercipienda punctis M, N <lb/>intercider; </s>
  <s xml:id="echoid-s2443" xml:space="preserve">puta ad φ. </s>
  <s xml:id="echoid-s2444" xml:space="preserve">quare QB quarta proportionalis erit in ratione <lb/>γ Qad Q φ; </s>
  <s xml:id="echoid-s2445" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2446" xml:space="preserve">eſt γ Q. </s>
  <s xml:id="echoid-s2447" xml:space="preserve">Qφ &amp;</s>
  <s xml:id="echoid-s2448" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2449" xml:space="preserve">(γ Q. </s>
  <s xml:id="echoid-s2450" xml:space="preserve">QN.) </s>
  <s xml:id="echoid-s2451" xml:space="preserve">:</s>
  <s xml:id="echoid-s2452" xml:space="preserve">: YP. </s>
  <s xml:id="echoid-s2453" xml:space="preserve">PN . </s>
  <s xml:id="echoid-s2454" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s2455" xml:space="preserve">
<pb o="44" file="0062" n="62" rhead=""/>
ſimplicibus triplicatas ſubſtituendo rationes, eſt γ Q. </s>
  <s xml:id="echoid-s2456" xml:space="preserve">QB &amp;</s>
  <s xml:id="echoid-s2457" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2458" xml:space="preserve">YP . <lb/></s>
  <s xml:id="echoid-s2459" xml:space="preserve">PB. </s>
  <s xml:id="echoid-s2460" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2461" xml:space="preserve">his æquales rationes adjungendo eſt QN. </s>
  <s xml:id="echoid-s2462" xml:space="preserve">γ Q + γ Q. </s>
  <s xml:id="echoid-s2463" xml:space="preserve">QB&amp;</s>
  <s xml:id="echoid-s2464" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2465" xml:space="preserve"><lb/>PN . </s>
  <s xml:id="echoid-s2466" xml:space="preserve">YP + YP. </s>
  <s xml:id="echoid-s2467" xml:space="preserve">PB; </s>
  <s xml:id="echoid-s2468" xml:space="preserve">hoc eſt QN. </s>
  <s xml:id="echoid-s2469" xml:space="preserve">QB &amp;</s>
  <s xml:id="echoid-s2470" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2471" xml:space="preserve">PN. </s>
  <s xml:id="echoid-s2472" xml:space="preserve">PB. </s>
  <s xml:id="echoid-s2473" xml:space="preserve">compo-<lb/>nendóque BN. </s>
  <s xml:id="echoid-s2474" xml:space="preserve">QB &amp;</s>
  <s xml:id="echoid-s2475" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2476" xml:space="preserve">BN. </s>
  <s xml:id="echoid-s2477" xml:space="preserve">P B. </s>
  <s xml:id="echoid-s2478" xml:space="preserve">ergo QB &amp;</s>
  <s xml:id="echoid-s2479" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s2480" xml:space="preserve">PB. </s>
  <s xml:id="echoid-s2481" xml:space="preserve">unde rurſus <lb/>liquet rectam X Qipſis AB; </s>
  <s xml:id="echoid-s2482" xml:space="preserve">ZP interjacere: </s>
  <s xml:id="echoid-s2483" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s2484" xml:space="preserve">E. </s>
  <s xml:id="echoid-s2485" xml:space="preserve">D.</s>
  <s xml:id="echoid-s2486" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2487" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s2488" xml:space="preserve">Conſimili prorſus argumentatione conſtabit obliquio-<lb/>rum incidentium refractos ultra punctum Z ipſam KN interſe-<lb/>care.</s>
  <s xml:id="echoid-s2489" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2490" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s2491" xml:space="preserve">Quinimò rurſus exertiùs apparet non niſi binos refractos <lb/>in eodem puncto convenire.</s>
  <s xml:id="echoid-s2492" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2493" xml:space="preserve">XI X. </s>
  <s xml:id="echoid-s2494" xml:space="preserve">Addo cum ipſo KN concurrentes refractos circa punctum <lb/>Z conglomerari, præſertim illos, qui ad partes F (obliquius) inci-<lb/>dentes pertinent.</s>
  <s xml:id="echoid-s2495" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2496" xml:space="preserve">Nam accipiantur, exempli causâ, ſibi pares N S, ST ; </s>
  <s xml:id="echoid-s2497" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2498" xml:space="preserve">ſit BP <lb/>
<anchor type="note" xlink:label="note-0062-01a" xlink:href="note-0062-01"/>
quarta proportionalis in ratione YB ad BN; </s>
  <s xml:id="echoid-s2499" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2500" xml:space="preserve">BQ quarta proportionalis <lb/>in ratione YB ad BS; </s>
  <s xml:id="echoid-s2501" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2502" xml:space="preserve">BR itidem quarta talis in ratione YB ad <lb/>BT; </s>
  <s xml:id="echoid-s2503" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2504" xml:space="preserve">à punctis P, Q, R erectæ perpendiculares ipſam NKZ <lb/>ſecent in Z, X, &amp; </s>
  <s xml:id="echoid-s2505" xml:space="preserve">V. </s>
  <s xml:id="echoid-s2506" xml:space="preserve">patet (è mox oſtenſis) omnium ſpatio NS <lb/>incidentium refractos cum NK concurrere intra ZX; </s>
  <s xml:id="echoid-s2507" xml:space="preserve">nec non omnes <lb/>ipſi ST incidentium refractos intra XV cum eodem convenire. </s>
  <s xml:id="echoid-s2508" xml:space="preserve">por-<lb/>rò rectæ BP , BQ, BR ſe habent invicem ut _Cubi_ rectarum BN, <lb/>BS, BT, ( vel ſunt in ipſarum BN, BS, BT _ratione triplicata_ : </s>
  <s xml:id="echoid-s2509" xml:space="preserve">nam <lb/>BP. </s>
  <s xml:id="echoid-s2510" xml:space="preserve">YB :</s>
  <s xml:id="echoid-s2511" xml:space="preserve">: BN cub. </s>
  <s xml:id="echoid-s2512" xml:space="preserve">BY cub. </s>
  <s xml:id="echoid-s2513" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2514" xml:space="preserve">YB . </s>
  <s xml:id="echoid-s2515" xml:space="preserve">BQ :</s>
  <s xml:id="echoid-s2516" xml:space="preserve">: YB cub. </s>
  <s xml:id="echoid-s2517" xml:space="preserve">BS cub. <lb/></s>
  <s xml:id="echoid-s2518" xml:space="preserve">adeóque ex æquo BP. </s>
  <s xml:id="echoid-s2519" xml:space="preserve">BQ :</s>
  <s xml:id="echoid-s2520" xml:space="preserve">: BN cub . </s>
  <s xml:id="echoid-s2521" xml:space="preserve">BS cub. </s>
  <s xml:id="echoid-s2522" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2523" xml:space="preserve">conſimili ratione <lb/>BP. </s>
  <s xml:id="echoid-s2524" xml:space="preserve">BR :</s>
  <s xml:id="echoid-s2525" xml:space="preserve">: BN cub. </s>
  <s xml:id="echoid-s2526" xml:space="preserve">BT cub.) </s>
  <s xml:id="echoid-s2527" xml:space="preserve">undè faciiè monſtrabitur eſſe PQ <lb/>multo minorem quàm QR; </s>
  <s xml:id="echoid-s2528" xml:space="preserve">xel ZX quàm XV (verbis parco multis <lb/>in re ſatis manifeſta) . </s>
  <s xml:id="echoid-s2529" xml:space="preserve">quare dicti refracti circa punctum Z ſpiſſiùs <lb/>ipſum NK decuſſabunt.</s>
  <s xml:id="echoid-s2530" xml:space="preserve"/>
</p>
<div xml:id="echoid-div67" type="float" level="2" n="10">
<note position="left" xlink:label="note-0062-01" xlink:href="note-0062-01a" xml:space="preserve">Fig. 61.</note>
</div>
<p>
  <s xml:id="echoid-s2531" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s2532" xml:space="preserve">Ex his demùm conficitur omnibus bene trutinatis, oculo <lb/>(O) centrum habenti in refracto NK uſpiam conſtituto, puncti A <lb/>imaginem ad ipſum conditione præditum toties inſinuatâ punctum Z <lb/>conſiſtere. </s>
  <s xml:id="echoid-s2533" xml:space="preserve">Sit enim CD pupillæ (in plano ABC jacens) _Diameter;_ <lb/></s>
  <s xml:id="echoid-s2534" xml:space="preserve">_axi Optico_ KN perpendicularis; </s>
  <s xml:id="echoid-s2535" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2536" xml:space="preserve">per ejus extrema C, D tranſeant <lb/>refracti IM LR ipſi KN occurrentes punctis X, V. </s>
  <s xml:id="echoid-s2537" xml:space="preserve">ex oſtenſis patet <lb/>omnium intra ſpatium MN incidentium radiorum refractos intra ten-<lb/>minos ZX principalem refractum interſecare; </s>
  <s xml:id="echoid-s2538" xml:space="preserve">neque non omnes ad
<pb o="45" file="0063" n="63" rhead=""/>
ſpatium NR pertinentes intra ZV eidem occurrere; </s>
  <s xml:id="echoid-s2539" xml:space="preserve">quinetiam nulli-<lb/>us citra punctum M, vel ultra R incidentis refractum (ſeu nullum <lb/>citra X, vel ultra V cum ipſo KN concurrentem) oculum ingredi <lb/>poſſe. </s>
  <s xml:id="echoid-s2540" xml:space="preserve">quare ſaltem imago conſiſtet intra terminos VX; </s>
  <s xml:id="echoid-s2541" xml:space="preserve">ſiquidem <lb/>aliunde qui videntur emanare _Radii_ nihil quicquam ad viſionem con-<lb/>ferent, aut ad eam ullatenus pertinebunt. </s>
  <s xml:id="echoid-s2542" xml:space="preserve">cæterum quoniam ab VX <lb/>procedentium (apparenter, inquam, procedentium) reciſſimi, vel <lb/>axi propriores velut ab ipſo Z procedere videntur (ſeu à loco qui <lb/>circa ipſum) ipsíque proinde validiùs afficiunt oculum, &amp; </s>
  <s xml:id="echoid-s2543" xml:space="preserve">ab eo <lb/>fa<unsure/>ciliùs adunari, recolligíque poſſunt; </s>
  <s xml:id="echoid-s2544" xml:space="preserve">cùm &amp; </s>
  <s xml:id="echoid-s2545" xml:space="preserve">ii præ cæteris confer-<lb/>tim irruant (illi ſaltem qui ad partes NR, ) quia denique propter <lb/>anguſtiam pupillæ ſpatium VX haud ita magnum exiſtit; </s>
  <s xml:id="echoid-s2546" xml:space="preserve">cùm, in-<lb/>guam, hæcità ſe habeant, omnino rationi conſentaneum eſt, dictam <lb/>imaginem circa punctum Z verſari; </s>
  <s xml:id="echoid-s2547" xml:space="preserve">nec alias arbitror excogitari poſſe <lb/>veriſimiles cauſas, quæ ſitum ejus determinent. </s>
  <s xml:id="echoid-s2548" xml:space="preserve">_Alhazenw_ quidem, <lb/>&amp; </s>
  <s xml:id="echoid-s2549" xml:space="preserve">poſt eum pleraque cohors _Opticorum_ ipſam ad punctum K, ubi <lb/>principalis refractus perpendicularem AB decuſſat, conſtituit; </s>
  <s xml:id="echoid-s2550" xml:space="preserve">ve-<lb/>
<anchor type="note" xlink:label="note-0063-01a" xlink:href="note-0063-01"/>
rùm haud ullam rei natura cauſam ſuggerit, cur inibi ſtatuatur. </s>
  <s xml:id="echoid-s2551" xml:space="preserve">unicus <lb/>enim (niſi ſaltem pupilla perpendicularem ipſam AB comprehendat, <lb/>oculuſque valdè ſit ei propinquus) per illud punctum means radius, <lb/>afficiendo viſui minimè ſuffecturus, ingredietur oculum; </s>
  <s xml:id="echoid-s2552" xml:space="preserve">eadémque <lb/>punctorum intra KX ipſi K adjacentium eſt ratio; </s>
  <s xml:id="echoid-s2553" xml:space="preserve">nullus ſiquidem <lb/>ea permeans refractus oculum attingit; </s>
  <s xml:id="echoid-s2554" xml:space="preserve">nil itaque ſubeſt cauſæ cur <lb/>punctum A circa K appareat. </s>
  <s xml:id="echoid-s2555" xml:space="preserve">quin adhuc à vero magis aberrat, *qui <lb/>
<anchor type="note" xlink:label="note-0063-02a" xlink:href="note-0063-02"/>
faciens NH æ qualem ipſi NA puncto H affigit imaginem (huc, opi-<lb/>nior, impulſus quia taliter in reflectione ſe rem habere perſpexit; <lb/></s>
  <s xml:id="echoid-s2556" xml:space="preserve">cui ſimilis cauſa nì fallor _Euclidem, Alhazenum, Stevinum_ (quan-<lb/>quam ipſos in diverſum euntes) horúmque ſequaces, in _Catoptricis_, <lb/>in errorem egit; </s>
  <s xml:id="echoid-s2557" xml:space="preserve">prout uſu non rarò venit _analogias haud bene_ <lb/>_fundatas,_ indiſtinctéque perceptas mortalibus imponere; </s>
  <s xml:id="echoid-s2558" xml:space="preserve">ſed utcunque <lb/>quod dixi magìs iſta ſententia abhorret à ratione. </s>
  <s xml:id="echoid-s2559" xml:space="preserve">) nullus enim in <lb/>primo caſu refractorum concurſus fit infra angulum ABF; </s>
  <s xml:id="echoid-s2560" xml:space="preserve">nullus extra <lb/>illum in ſecundo; </s>
  <s xml:id="echoid-s2561" xml:space="preserve">proindèque fortiùs hanc quæ objecimus, quàm <lb/>priorem _Alhazeni_ percellunt ſententiam. </s>
  <s xml:id="echoid-s2562" xml:space="preserve">addo quòd ſimul utraque, <lb/>ſed præſertim hæc, multiplici refragatur experientiæ, multiplicíque <lb/>ratiocinio; </s>
  <s xml:id="echoid-s2563" xml:space="preserve">pariter enim ſe res habere debuit in _Catoptricis_, ut &amp; </s>
  <s xml:id="echoid-s2564" xml:space="preserve"><lb/>in _Dioptricis circularibus_; </s>
  <s xml:id="echoid-s2565" xml:space="preserve">id quòd maniſeſtè, longéque ſecus tam <lb/>ab experientiâ, quàm à ratione compertum eſt. </s>
  <s xml:id="echoid-s2566" xml:space="preserve">quin hanc abunde <lb/>ſubvertit ac peſſum dat quod ſupra propoſuimus experimentum, nemi-<lb/>ni non obvium; </s>
  <s xml:id="echoid-s2567" xml:space="preserve">quod nempepunctum A, oculo in ipſo perpendicu.</s>
  <s xml:id="echoid-s2568" xml:space="preserve">
<pb o="46" file="0064" n="64" rhead=""/>
lari AB conſtituto, non in ſuo loco (quod juxta dictam ſententiam <lb/>oportuit) aſt pro mediorum diverſitate, (perquam ſenſibili inter vallo) <lb/>citeriùs adſpectatur, aut ulteriùs. </s>
  <s xml:id="echoid-s2569" xml:space="preserve">Sed effatum hoc noſtrum (eíque <lb/>quoad reliquos in Catoptricis, Dioptricíſque caſus ſimiles conſona) <lb/>tametſi novitium, &amp; </s>
  <s xml:id="echoid-s2570" xml:space="preserve">nullâ quod ſciam hactenus auctoritate fultum, <lb/>cùm forſan expoſitum dilucidiùs, tum penitiſſimè dabimus confir-<lb/>matum, ſi quando nos de imaginum natura, locóque ſpeciatim <lb/>evenerit diſſertare. </s>
  <s xml:id="echoid-s2571" xml:space="preserve">Mihi ſaltem videtur hæc Scientia quoad hanc <lb/>partem ſuam, certè palmariam (utì reperitur hactenus tractata) <lb/>perquam mutila, nè dicam admodùm vitioſa; </s>
  <s xml:id="echoid-s2572" xml:space="preserve">nec aliò ferè collima-<lb/>mus quàm ut aliquouſque ſuppleamus eam, ac ſanemus.</s>
  <s xml:id="echoid-s2573" xml:space="preserve"/>
</p>
<div xml:id="echoid-div68" type="float" level="2" n="11">
<note position="right" xlink:label="note-0063-01" xlink:href="note-0063-01a" xml:space="preserve">Fig. 62.</note>
<note position="right" xlink:label="note-0063-02" xlink:href="note-0063-02a" xml:space="preserve">_D. Hθbbiw._</note>
</div>
<p>
  <s xml:id="echoid-s2574" xml:space="preserve">XXI. </s>
  <s xml:id="echoid-s2575" xml:space="preserve">Proximè dictis confirmandis idoneum haud illepidum expe-<lb/>
<anchor type="note" xlink:label="note-0064-01a" xlink:href="note-0064-01"/>
rimentum interſeremus. </s>
  <s xml:id="echoid-s2576" xml:space="preserve">Aqueæ Superficiei RS (ſtagnanti, &amp; </s>
  <s xml:id="echoid-s2577" xml:space="preserve"><lb/>immotæ deſuper immineat objectum HG; </s>
  <s xml:id="echoid-s2578" xml:space="preserve">ejus autem punctum G ra-<lb/>dat perpendiculum EF (filum puta candidum, aut ſtylus, cui <lb/>plumbum F appenditur) videbitur itaque punctum G (oculo O) ex <lb/>reflectione in ipſà perpendiculari GB velut ad γ; </s>
  <s xml:id="echoid-s2579" xml:space="preserve">at perpendiculi <lb/>punctum F (admodum notabili diſtantiâ) citra lineam B γ aſpicitur <lb/>(velut ad φ) id quod ex ſententia noſtra factum oportuit; </s>
  <s xml:id="echoid-s2580" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2581" xml:space="preserve">_Albaze-_ <lb/>_ni_, ſequaciúmque doctrinam liquidò deſtruit.</s>
  <s xml:id="echoid-s2582" xml:space="preserve"/>
</p>
<div xml:id="echoid-div69" type="float" level="2" n="12">
<note position="left" xlink:label="note-0064-01" xlink:href="note-0064-01a" xml:space="preserve">Fig. 63.</note>
</div>
<p>
  <s xml:id="echoid-s2583" xml:space="preserve">XXII. </s>
  <s xml:id="echoid-s2584" xml:space="preserve">Subjicio tandem ex his comparere modum genuinam refra-<lb/>ctariam quam vocant (per quam nempe recta linea repræſentatur <lb/>in aquæ fundo conſpicua) lineam deſignandi; </s>
  <s xml:id="echoid-s2585" xml:space="preserve">cujus loco complures <lb/>(utique non eandem omnes, aſt aliam alii) Chimæram inani ſunt <lb/>operà proſequuti; </s>
  <s xml:id="echoid-s2586" xml:space="preserve">de quâ _Carteſius_ ipſe percontanti _Merſenno_ ſic <lb/>reſpondit: </s>
  <s xml:id="echoid-s2587" xml:space="preserve">“ Non poteſt facilè determinari qualem figuram linea viſa <lb/>
<anchor type="note" xlink:label="note-0064-02a" xlink:href="note-0064-02"/>
in fundo aquæ ſit habitura; </s>
  <s xml:id="echoid-s2588" xml:space="preserve">neque enim certus eſt aliquis ima-<lb/>ginis locus in reſlexis aut reſractis, quemadmodum ſibi vulgò per-<lb/>ſuaſerunt Optici. </s>
  <s xml:id="echoid-s2589" xml:space="preserve">lmò verò (tanti viri pace) cùm ſpeciale quodvis <lb/>objectum (per ejuſdem generis &amp; </s>
  <s xml:id="echoid-s2590" xml:space="preserve">eodem modo terminatum medium <lb/>aſpectabile) ſimilem conſtanter exhibeat ſpeciem ſui, ſimili ſitu diſpo-<lb/>ſitam, ſimili præditam figurâ; </s>
  <s xml:id="echoid-s2591" xml:space="preserve">non video quin ex parte rei certum <lb/>imago locum ſortiatur; </s>
  <s xml:id="echoid-s2592" xml:space="preserve">cujus certè (quoad illum qui præ manibus <lb/>eſt caſum) quotcunque puncta non difficilè poterunt è præcedentibus <lb/>determinari. </s>
  <s xml:id="echoid-s2593" xml:space="preserve">quinimò nullius non. </s>
  <s xml:id="echoid-s2594" xml:space="preserve">ex hujuſmodi planam ad ſuperfici-<lb/>em refractione ſubnaſcentis phænomeni (quoad ejus intelligo figuram) <lb/>cauſa verè, nì fallor, hinc &amp; </s>
  <s xml:id="echoid-s2595" xml:space="preserve">promptè poſſit aſſignari. </s>
  <s xml:id="echoid-s2596" xml:space="preserve">verùm hæc circa <lb/>planas ſuperficies dicta ſuſſicient; </s>
  <s xml:id="echoid-s2597" xml:space="preserve">ad curvas nos proximè conferemus. </s>
  <s xml:id="echoid-s2598" xml:space="preserve">‖</s>
</p>
<div xml:id="echoid-div70" type="float" level="2" n="13">
<note position="left" xlink:label="note-0064-02" xlink:href="note-0064-02a" xml:space="preserve">Tom. II. Epiſt. <lb/>73.</note>
</div>
<pb o="47" file="0065" n="65"/>
</div>
<div xml:id="echoid-div72" type="section" level="1" n="15">
<head xml:id="echoid-head18" xml:space="preserve"><emph style="sc">Lect</emph>. VI.</head>
<p>
  <s xml:id="echoid-s2599" xml:space="preserve">I. </s>
  <s xml:id="echoid-s2600" xml:space="preserve">ABſolutis iis, quæ radiis accidunt ad planam ſuperficiem in-<lb/>flexis (obſervatu quæ videbantur non indigna, cúmque: </s>
  <s xml:id="echoid-s2601" xml:space="preserve">prin-<lb/>cipiis noſtris cohærentia; </s>
  <s xml:id="echoid-s2602" xml:space="preserve">quæ denuò viam ſternebant, aut metho-<lb/>dum aperiebant ſequentibus) ad curvas jam gradum promovemus; <lb/></s>
  <s xml:id="echoid-s2603" xml:space="preserve">circa quas equidem cogitâram communia quædam delibare; </s>
  <s xml:id="echoid-s2604" xml:space="preserve">verùm <lb/>excuſſà re, tam exilem illam &amp; </s>
  <s xml:id="echoid-s2605" xml:space="preserve">abſtractam deprehendo, ſatius ut <lb/>exiſtimem actutùm ad particularia deſcendere. </s>
  <s xml:id="echoid-s2606" xml:space="preserve">curvarum utique prin-<lb/>cipem, &amp; </s>
  <s xml:id="echoid-s2607" xml:space="preserve">ad praxes Opticas longè paratiſſimam, Superficiem Sphæ-<lb/>ricam aggrediar è veſtigio. </s>
  <s xml:id="echoid-s2608" xml:space="preserve">pro qua tamen, ob cauſas pridem aſſigna-<lb/>tas, circulos ſubrogabo per oculi Sphæræque centra, pérque ſingula <lb/>radiantia puncta trajectos. </s>
  <s xml:id="echoid-s2609" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2610" xml:space="preserve">quoad hos _Catoptrica_ primo, _D@op-_ <lb/>_trica_ poſtmodum exequemur. </s>
  <s xml:id="echoid-s2611" xml:space="preserve">Ad illa.</s>
  <s xml:id="echoid-s2612" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2613" xml:space="preserve">II. </s>
  <s xml:id="echoid-s2614" xml:space="preserve">Præſternemus autem _λημμdηον_ unum vel alterum; </s>
  <s xml:id="echoid-s2615" xml:space="preserve">hoc impri-<lb/>mìs: </s>
  <s xml:id="echoid-s2616" xml:space="preserve">Incidenti@m circulo radiorum obliquior eſt, qui magis à centro <lb/>
<anchor type="note" xlink:label="note-0065-01a" xlink:href="note-0065-01"/>
diſtat; </s>
  <s xml:id="echoid-s2617" xml:space="preserve">vel qui minorem arcum (ſubſemicircularem) ſubtendit. </s>
  <s xml:id="echoid-s2618" xml:space="preserve">ſci-<lb/>licet obliquiùs incidit recta QRS, quàm recta MNP. </s>
  <s xml:id="echoid-s2619" xml:space="preserve">Nam à <lb/>centro C ducantur CN, CP; </s>
  <s xml:id="echoid-s2620" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2621" xml:space="preserve">CR; </s>
  <s xml:id="echoid-s2622" xml:space="preserve">CS. </s>
  <s xml:id="echoid-s2623" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2624" xml:space="preserve">quoniam angulus <lb/>RCS angulo NCP (hypotheſi nimirum inſiſtendo) minor eſt; <lb/></s>
  <s xml:id="echoid-s2625" xml:space="preserve">pater reliquos CRS, CSR reliquos CN, P, CPN (cùm junctim, <lb/>tum ſingulum ſingulo) majores eſſe. </s>
  <s xml:id="echoid-s2626" xml:space="preserve">Cùm itaque Semidiametxi <lb/>CR, CN, circumferentiæ perpendiculares ſint; </s>
  <s xml:id="echoid-s2627" xml:space="preserve">omninò liquet pro-<lb/>poſitum.</s>
  <s xml:id="echoid-s2628" xml:space="preserve"/>
</p>
<div xml:id="echoid-div72" type="float" level="2" n="1">
<note position="right" xlink:label="note-0065-01" xlink:href="note-0065-01a" xml:space="preserve">Fig. 62.</note>
</div>
<p style="it">
  <s xml:id="echoid-s2629" xml:space="preserve">III. </s>
  <s xml:id="echoid-s2630" xml:space="preserve">Dato radio MN ad circulum incidenti congruum reflexum <lb/>reſignare.</s>
  <s xml:id="echoid-s2631" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 63.</note>
<p>
  <s xml:id="echoid-s2632" xml:space="preserve">Variis modis huc facilè peragitur; </s>
  <s xml:id="echoid-s2633" xml:space="preserve">quorum nunc unum adhibere, <lb/>tunc alium ex uſu ſit. </s>
  <s xml:id="echoid-s2634" xml:space="preserve">nos unum aut alterum ex expeditoribus attinge-<lb/>mus. </s>
  <s xml:id="echoid-s2635" xml:space="preserve">i. </s>
  <s xml:id="echoid-s2636" xml:space="preserve">Incidens MN protrahatur ut circulum denuò ſecet in P. </s>
  <s xml:id="echoid-s2637" xml:space="preserve">; <lb/></s>
  <s xml:id="echoid-s2638" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2639" xml:space="preserve">ſumatur arcus N Π = NP; </s>
  <s xml:id="echoid-s2640" xml:space="preserve">erit connexa Π NH ipſius MNP <lb/>reflexus. </s>
  <s xml:id="echoid-s2641" xml:space="preserve">nam à centro C connexâ CN, manifeſtum eſt angulum
<pb o="48" file="0066" n="66" rhead=""/>
CN Π, angulo CNP æquari. </s>
  <s xml:id="echoid-s2642" xml:space="preserve">z 2. </s>
  <s xml:id="echoid-s2643" xml:space="preserve">Accepto quovis in NM puncto <lb/>(puta M) centro C per M deſcribatur circulus MQH; </s>
  <s xml:id="echoid-s2644" xml:space="preserve">item centro <lb/>
<anchor type="note" xlink:label="note-0066-01a" xlink:href="note-0066-01"/>
N per M deſcribatur circulus MRH, qui priorem MQH ſecet <lb/>in H; </s>
  <s xml:id="echoid-s2645" xml:space="preserve">erit HN Π reflexus ipſius MN P. </s>
  <s xml:id="echoid-s2646" xml:space="preserve">Etenim connexis CM, <lb/>CH; </s>
  <s xml:id="echoid-s2647" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2648" xml:space="preserve">NM, NH, ex conſtructione liquet triangula CMN, CHN, <lb/>invicem æquilatera fore; </s>
  <s xml:id="echoid-s2649" xml:space="preserve">proindéque angulos CNM, CNH (&amp; </s>
  <s xml:id="echoid-s2650" xml:space="preserve"><lb/>indè relìquos MNR, HNR) æquari 3. </s>
  <s xml:id="echoid-s2651" xml:space="preserve">protenſà CNR, à quovis <lb/>in MN puncto, puta M ducatur MG ad CR perpendicularis, &amp; </s>
  <s xml:id="echoid-s2652" xml:space="preserve"><lb/>in hac producta ſumatur GH = GM; </s>
  <s xml:id="echoid-s2653" xml:space="preserve">erit conjuncta HN Π iterum <lb/>reflexus. </s>
  <s xml:id="echoid-s2654" xml:space="preserve">Nam connexis NH, NM patet angulos GNM, GNH <lb/>æquari. </s>
  <s xml:id="echoid-s2655" xml:space="preserve">verum hi modi ſufficiunt huic conficiendo perfacili negotio.</s>
  <s xml:id="echoid-s2656" xml:space="preserve"/>
</p>
<div xml:id="echoid-div73" type="float" level="2" n="2">
<note position="left" xlink:label="note-0066-01" xlink:href="note-0066-01a" xml:space="preserve">Fig. 63.</note>
</div>
<p>
  <s xml:id="echoid-s2657" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s2658" xml:space="preserve">Nocetur ſi fuerit HN P reflexus ipſius MN P fore N Π = <lb/>N P.</s>
  <s xml:id="echoid-s2659" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2660" xml:space="preserve">V. </s>
  <s xml:id="echoid-s2661" xml:space="preserve">Diſpiciamus jam primò quid ex hujuſmodi reflectione contingat <lb/>puncto ab infinitâ quo ad ſenſum diſtantiâ radianti, ſeu parallelos <lb/>projicienti radios. </s>
  <s xml:id="echoid-s2662" xml:space="preserve">quorſum, per circuli reflectentis centrum C <lb/>protendatur indefinitè recta ABC (hoc autem in ſequentibus evitandæ <lb/>repetitioni perpetuò factum intelligatur; </s>
  <s xml:id="echoid-s2663" xml:space="preserve">quin ejuſmodi recta nomi-<lb/>netur axis; </s>
  <s xml:id="echoid-s2664" xml:space="preserve">hîc _Speculi,_ poſteà _Diapbani_) biſecetur autem Semidiame-<lb/>ter CB in Z; </s>
  <s xml:id="echoid-s2665" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2666" xml:space="preserve">per Z tranſeat recta ZY ad CB perpendicularis, <lb/>indeſinitéque protenſa ; </s>
  <s xml:id="echoid-s2667" xml:space="preserve">tum quilibet incidat axi parallelus radius <lb/>MN P ad N; </s>
  <s xml:id="echoid-s2668" xml:space="preserve">(convexo circuli nil refert, an cavo; </s>
  <s xml:id="echoid-s2669" xml:space="preserve">nam in utroque <lb/>caſu reflexus quoad directionem idem erit; </s>
  <s xml:id="echoid-s2670" xml:space="preserve">vel ejus qui in hoc, iſte qui <lb/>in illo productus erit) connexáque CN ipſam ZY interſecet in V; <lb/></s>
  <s xml:id="echoid-s2671" xml:space="preserve">ſiátque CK = CV; </s>
  <s xml:id="echoid-s2672" xml:space="preserve">ducatúrque NK; </s>
  <s xml:id="echoid-s2673" xml:space="preserve">erit NK ipſius MN reflex-<lb/>us (vel reflexi productus) Nam ducatur NQ ad CB perpendicu-<lb/>laris, &amp; </s>
  <s xml:id="echoid-s2674" xml:space="preserve">connectatur CP. </s>
  <s xml:id="echoid-s2675" xml:space="preserve">éſtque CZ . </s>
  <s xml:id="echoid-s2676" xml:space="preserve">CK :</s>
  <s xml:id="echoid-s2677" xml:space="preserve">: (CZ. </s>
  <s xml:id="echoid-s2678" xml:space="preserve">CV :</s>
  <s xml:id="echoid-s2679" xml:space="preserve">: ) <lb/>CQ. </s>
  <s xml:id="echoid-s2680" xml:space="preserve">CN . </s>
  <s xml:id="echoid-s2681" xml:space="preserve">quapropter antecedentes duplicando CN . </s>
  <s xml:id="echoid-s2682" xml:space="preserve">CK :</s>
  <s xml:id="echoid-s2683" xml:space="preserve">: <lb/>PN. </s>
  <s xml:id="echoid-s2684" xml:space="preserve">CN. </s>
  <s xml:id="echoid-s2685" xml:space="preserve">item angulus KCN æquatur alterno CNP. </s>
  <s xml:id="echoid-s2686" xml:space="preserve">ergò tri-<lb/>angula CKN, NCP ſimilia ſunt; </s>
  <s xml:id="echoid-s2687" xml:space="preserve">adeoque KN = KC. </s>
  <s xml:id="echoid-s2688" xml:space="preserve">igitur è <lb/>ſuprà generatim oſtenſis patet fore KN, ipſius MN reflexum.</s>
  <s xml:id="echoid-s2689" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2690" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s2691" xml:space="preserve">Hinc particularis emergit methodus hujuſmodi quotcunque <lb/>reflexos quàm expeditiſſime deſignandi; </s>
  <s xml:id="echoid-s2692" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s2693" xml:space="preserve">ipſorum erga ſe ra-<lb/>tiones ac reſpectus; </s>
  <s xml:id="echoid-s2694" xml:space="preserve">nec non pleraque primaria _Symptomata_ facilè <lb/>diluceſcunt; </s>
  <s xml:id="echoid-s2695" xml:space="preserve">corollariis nempe ſubjectis comprehenſa.</s>
  <s xml:id="echoid-s2696" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2697" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s2698" xml:space="preserve">I. </s>
  <s xml:id="echoid-s2699" xml:space="preserve">Patet punctum Z, Semidiametrum CB biſecans, eſſe
<pb o="49" file="0067" n="67" rhead=""/>
metam infra quam nullus reflexus axem ſecat (vel perpendicularis <lb/>iqſius reflexum BZ ad Z terminari). </s>
  <s xml:id="echoid-s2700" xml:space="preserve">quia ſemper Cv &amp;</s>
  <s xml:id="echoid-s2701" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2702" xml:space="preserve">CZ; <lb/></s>
  <s xml:id="echoid-s2703" xml:space="preserve">adeóque CK &amp;</s>
  <s xml:id="echoid-s2704" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2705" xml:space="preserve">CZ.</s>
  <s xml:id="echoid-s2706" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2707" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s2708" xml:space="preserve">2. </s>
  <s xml:id="echoid-s2709" xml:space="preserve">Patet eſſe KN = KC. <lb/></s>
  <s xml:id="echoid-s2710" xml:space="preserve">3. </s>
  <s xml:id="echoid-s2711" xml:space="preserve">Patet fore PN (2 CQ), CN, CK {.</s>
  <s xml:id="echoid-s2712" xml:space="preserve">./.</s>
  <s xml:id="echoid-s2713" xml:space="preserve">.}</s>
</p>
<p>
  <s xml:id="echoid-s2714" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s2715" xml:space="preserve">4. </s>
  <s xml:id="echoid-s2716" xml:space="preserve">Ductâ tangente BT, productâque CNE, patet ſecan-<lb/>tem CE diftantiæ CK duplam eſſe; </s>
  <s xml:id="echoid-s2717" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2718" xml:space="preserve">EN = 2 KZ.</s>
  <s xml:id="echoid-s2719" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2720" xml:space="preserve">X. </s>
  <s xml:id="echoid-s2721" xml:space="preserve">5. </s>
  <s xml:id="echoid-s2722" xml:space="preserve">Manifeſtum eſt incidentis ad F (hoc eſt ad diſtantiam 60 <lb/>
<anchor type="note" xlink:label="note-0067-01a" xlink:href="note-0067-01"/>
graduum à vertice) reflexum per verticem B tranſire; </s>
  <s xml:id="echoid-s2723" xml:space="preserve">proindéque <lb/>reflexos omnium intra BF incidentium axem intra ſpacium BZ decuſſa-<lb/>re; </s>
  <s xml:id="echoid-s2724" xml:space="preserve">ſed omnes _extra_ BF reflexos ultra B cum eo convenire.</s>
  <s xml:id="echoid-s2725" xml:space="preserve"/>
</p>
<div xml:id="echoid-div74" type="float" level="2" n="3">
<note position="right" xlink:label="note-0067-01" xlink:href="note-0067-01a" xml:space="preserve">Fig. 64.</note>
</div>
<p>
  <s xml:id="echoid-s2726" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s2727" xml:space="preserve">6. </s>
  <s xml:id="echoid-s2728" xml:space="preserve">Perſpicuum eſt duorum hujuſmodi quorumvis ad eaſdem <lb/>
<anchor type="note" xlink:label="note-0067-02a" xlink:href="note-0067-02"/>
axis partes incidentium (ut ipſorum MNP, QRS) reflexos (ut <lb/>GNK, HRL,) productos ſe prius decuſſare, quàm axem. </s>
  <s xml:id="echoid-s2729" xml:space="preserve">Nam, <lb/>ductis CR, CN, eſt C ρ &amp;</s>
  <s xml:id="echoid-s2730" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2731" xml:space="preserve">Cv, adeóque CL &amp;</s>
  <s xml:id="echoid-s2732" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2733" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s2734" xml:space="preserve">unde ne-<lb/>ceſſariò rectæ NK, RL, ſe decuſſabunt, puta ad X.</s>
  <s xml:id="echoid-s2735" xml:space="preserve"/>
</p>
<div xml:id="echoid-div75" type="float" level="2" n="4">
<note position="right" xlink:label="note-0067-02" xlink:href="note-0067-02a" xml:space="preserve">Fig. 65.</note>
</div>
<p>
  <s xml:id="echoid-s2736" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s2737" xml:space="preserve">Hinc ipſi convexis partibus incidentium reflexi, NG, RH, <lb/>antrorſum procurrentes divergunt; </s>
  <s xml:id="echoid-s2738" xml:space="preserve">adeóque nunquam uno plures <lb/>idem oculi centrum permeant. </s>
  <s xml:id="echoid-s2739" xml:space="preserve">unde ſpeculum convexum unicam lon-<lb/>ginqui radiantis imaginem reddit.</s>
  <s xml:id="echoid-s2740" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2741" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s2742" xml:space="preserve">7. </s>
  <s xml:id="echoid-s2743" xml:space="preserve">Notetur autem angulum GXR (vel KXL ) à duobus <lb/>reflexis comprehenſum æquare duplum angulum NCR (hoc eſt <lb/>duplum exceſſum angulorum incidentiæ) . </s>
  <s xml:id="echoid-s2744" xml:space="preserve">Nam ang. </s>
  <s xml:id="echoid-s2745" xml:space="preserve">KXL = ang. <lb/></s>
  <s xml:id="echoid-s2746" xml:space="preserve">ALR - ang. </s>
  <s xml:id="echoid-s2747" xml:space="preserve">AKN = 2 ang. </s>
  <s xml:id="echoid-s2748" xml:space="preserve">ACR - 2 ang. </s>
  <s xml:id="echoid-s2749" xml:space="preserve">ACN = 2 ang. </s>
  <s xml:id="echoid-s2750" xml:space="preserve"><lb/>NCR.</s>
  <s xml:id="echoid-s2751" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2752" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s2753" xml:space="preserve">Pro Sequentibus hujuſmodi _Lemma_ proponemus: </s>
  <s xml:id="echoid-s2754" xml:space="preserve">In trian-<lb/>gulo quopiam ABC recta AD biſecet angulum BAC; </s>
  <s xml:id="echoid-s2755" xml:space="preserve">dico fore <lb/>
<anchor type="note" xlink:label="note-0067-03a" xlink:href="note-0067-03"/>
AB + AC &amp;</s>
  <s xml:id="echoid-s2756" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2757" xml:space="preserve">2 A D.</s>
  <s xml:id="echoid-s2758" xml:space="preserve"/>
</p>
<div xml:id="echoid-div76" type="float" level="2" n="5">
<note position="right" xlink:label="note-0067-03" xlink:href="note-0067-03a" xml:space="preserve">Fig. 66.</note>
</div>
<p>
  <s xml:id="echoid-s2759" xml:space="preserve">In _Iſoſcele_ res clara eſt; </s>
  <s xml:id="echoid-s2760" xml:space="preserve">in alio proinde ſit AC &amp;</s>
  <s xml:id="echoid-s2761" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2762" xml:space="preserve">AB; </s>
  <s xml:id="echoid-s2763" xml:space="preserve">centró-<lb/>que A per B ducatur circulus BXY ſecans ipſam. </s>
  <s xml:id="echoid-s2764" xml:space="preserve">AD in X, &amp; </s>
  <s xml:id="echoid-s2765" xml:space="preserve">AC <lb/>in Y. </s>
  <s xml:id="echoid-s2766" xml:space="preserve">Subtenſa BX ducatur, ipſamque AC ſecet in V; </s>
  <s xml:id="echoid-s2767" xml:space="preserve">fiátque <lb/>VT ad AD parallela. </s>
  <s xml:id="echoid-s2768" xml:space="preserve">denuo ſubtenſa XY connectatur. </s>
  <s xml:id="echoid-s2769" xml:space="preserve">Et quoni-<lb/>am ang. </s>
  <s xml:id="echoid-s2770" xml:space="preserve">XVC major eſt angulo XYV, vel angulo BXD, vel ipſo
<pb o="50" file="0068" n="68" rhead=""/>
BVT, patet rectam VT angulum XVC ſecare. </s>
  <s xml:id="echoid-s2771" xml:space="preserve">item ob angulum <lb/>XYV obtuſum, eſt XV &amp;</s>
  <s xml:id="echoid-s2772" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2773" xml:space="preserve">XY = BX. </s>
  <s xml:id="echoid-s2774" xml:space="preserve">ergo BV &amp;</s>
  <s xml:id="echoid-s2775" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2776" xml:space="preserve">2 BX; </s>
  <s xml:id="echoid-s2777" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2778" xml:space="preserve"><lb/>VT &amp;</s>
  <s xml:id="echoid-s2779" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2780" xml:space="preserve">2 XD. </s>
  <s xml:id="echoid-s2781" xml:space="preserve">Verùm ang. </s>
  <s xml:id="echoid-s2782" xml:space="preserve">VTC (major ipſo TVB, vel ipſo <lb/>DXB) eſt obtuſus; </s>
  <s xml:id="echoid-s2783" xml:space="preserve">adeóque VC &amp;</s>
  <s xml:id="echoid-s2784" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2785" xml:space="preserve">VT; </s>
  <s xml:id="echoid-s2786" xml:space="preserve">itàque magìs YC &amp;</s>
  <s xml:id="echoid-s2787" xml:space="preserve">gt; <lb/></s>
  <s xml:id="echoid-s2788" xml:space="preserve">2 X D. </s>
  <s xml:id="echoid-s2789" xml:space="preserve">ergò AB + AY + YC &amp;</s>
  <s xml:id="echoid-s2790" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2791" xml:space="preserve">2 AX + 2 XD. </s>
  <s xml:id="echoid-s2792" xml:space="preserve">hoc eſt <lb/>AB + AC &amp;</s>
  <s xml:id="echoid-s2793" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2794" xml:space="preserve">2 AD : </s>
  <s xml:id="echoid-s2795" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s2796" xml:space="preserve">E. </s>
  <s xml:id="echoid-s2797" xml:space="preserve">D.</s>
  <s xml:id="echoid-s2798" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2799" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s2800" xml:space="preserve">Quò paralleli radii rectiùs (vel axi propinquiùs) incidunt, <lb/>eò reflexorum concurſus ad axem ſibi viciniores ſunt.</s>
  <s xml:id="echoid-s2801" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2802" xml:space="preserve">Nempe ſumantur utcunque pares arcus NR, RX; </s>
  <s xml:id="echoid-s2803" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2804" xml:space="preserve">incidentium <lb/>
<anchor type="note" xlink:label="note-0068-01a" xlink:href="note-0068-01"/>
MN, QR, VX reflexi NK, RL, XM cum axe conveniant punctis <lb/>K, L, M, erit ML &amp;</s>
  <s xml:id="echoid-s2805" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2806" xml:space="preserve">LK. </s>
  <s xml:id="echoid-s2807" xml:space="preserve">Nam connexæ CN, CR, CX rectæ <lb/>ZY occurrant punctis v, ρ, ξ. </s>
  <s xml:id="echoid-s2808" xml:space="preserve">Eſt itaque (juxta _Lemma_ præcedens) <lb/>C ξ + Cv &amp;</s>
  <s xml:id="echoid-s2809" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2810" xml:space="preserve">2C ρ; </s>
  <s xml:id="echoid-s2811" xml:space="preserve">hoc eſt, CM + CK &amp;</s>
  <s xml:id="echoid-s2812" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2813" xml:space="preserve">2 CL. </s>
  <s xml:id="echoid-s2814" xml:space="preserve">quare CM <lb/>- CL &amp;</s>
  <s xml:id="echoid-s2815" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2816" xml:space="preserve">CL - CK; </s>
  <s xml:id="echoid-s2817" xml:space="preserve">hoceſt ML &amp;</s>
  <s xml:id="echoid-s2818" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2819" xml:space="preserve">LK: </s>
  <s xml:id="echoid-s2820" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s2821" xml:space="preserve">E. </s>
  <s xml:id="echoid-s2822" xml:space="preserve">D.</s>
  <s xml:id="echoid-s2823" xml:space="preserve"/>
</p>
<div xml:id="echoid-div77" type="float" level="2" n="6">
<note position="left" xlink:label="note-0068-01" xlink:href="note-0068-01a" xml:space="preserve">Fig. 67.</note>
</div>
<p>
  <s xml:id="echoid-s2824" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s2825" xml:space="preserve">Exhinc patet axi propinquam lucem ab hujuſmodi reflectione <lb/>magìs magíſque conſtipari; </s>
  <s xml:id="echoid-s2826" xml:space="preserve">maximè circa punctum Z, ubi perpendi-<lb/>cularis ipſius quaſi reflexus terminatur. </s>
  <s xml:id="echoid-s2827" xml:space="preserve">unde potiſſima conſtat ratio, <lb/>quare concavis à ſpeculis ad ſolem expoſitis<unsure/> circa punctum Z _Ignis_ <lb/>accenditur; </s>
  <s xml:id="echoid-s2828" xml:space="preserve">e<unsure/>nimverò condenſatior, ínque ſpacium arctius quaſi com-<lb/>preſſa lux validiorem exerit vim, ac efficaciam.</s>
  <s xml:id="echoid-s2829" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2830" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s2831" xml:space="preserve">Quinetiam ex his conſectatur, longinqui puncti imaginem <lb/>oculo in axe conſtitnta circa punctum Z conſiſtere. </s>
  <s xml:id="echoid-s2832" xml:space="preserve">Sit, inquam, <lb/>
<anchor type="note" xlink:label="note-0068-02a" xlink:href="note-0068-02"/>
BCO axis Opticns; </s>
  <s xml:id="echoid-s2833" xml:space="preserve">oculíque dìameter D δ (in plana nempe cir-<lb/>culi propoſiti ſita) hujus autem extrema permeent reflexi NKD, <lb/>VK δ (ad incidentes MN P, μ ν Π pertinentes) . </s>
  <s xml:id="echoid-s2834" xml:space="preserve">jam abunde mani-<lb/>feſtum eſt imaginem conſpicuam intra KZ ſpatium verſari. </s>
  <s xml:id="echoid-s2835" xml:space="preserve">Nam <lb/>alterius cujuſvis hinc, vel indè cadentis reflexus (ſeu ipſius R S, vel <lb/>ρ σ) oculum omnino transgredietur, adeóque nihil quicquam ad viſio-<lb/>nem ipſam, vel ad ejus quemcunque modum determinandum conferet; <lb/></s>
  <s xml:id="echoid-s2836" xml:space="preserve">id autem omne meritó tribuetur radiorum intra peripheriam NV in-<lb/>cidentium reflexis; </s>
  <s xml:id="echoid-s2837" xml:space="preserve">qui ſcilicet oculum ingredientes ſuo quiſque modo <lb/>viſum aliquatenus afficiant. </s>
  <s xml:id="echoid-s2838" xml:space="preserve">quoniam tamen ex his, qui propiores <lb/>axi rectiùs incidunt oculo, magíſque pollent idcircò; </s>
  <s xml:id="echoid-s2839" xml:space="preserve">nec non iidem <lb/>proptergà faciliùs ad unum in oculo punctum recolliguntur; </s>
  <s xml:id="echoid-s2840" xml:space="preserve">præ cæte-<lb/>ris etiam illi catervatim ingruunt; </s>
  <s xml:id="echoid-s2841" xml:space="preserve">rationi conſonum eſt iſthic præſer-<lb/>tim imaginem conſiſtere; </s>
  <s xml:id="echoid-s2842" xml:space="preserve">ſiquidem velut ab eo plures, ac efficaciſſimi <lb/>radii videbuntur cmanare, Subjicio, propter admodum exiguam pu-
<pb o="51" file="0069" n="69" rhead=""/>
pillæ latitudinem, ipſum ſpatium KZ non ità magnum eſſe; </s>
  <s xml:id="echoid-s2843" xml:space="preserve">quin inſtat <lb/>_Puncti_ poſſit cenſeri. </s>
  <s xml:id="echoid-s2844" xml:space="preserve">Quibus expenſis luculentè conſtare videtur pro-<lb/>poſitum.</s>
  <s xml:id="echoid-s2845" xml:space="preserve"/>
</p>
<div xml:id="echoid-div78" type="float" level="2" n="7">
<note position="left" xlink:label="note-0068-02" xlink:href="note-0068-02a" xml:space="preserve">Fig. 68.</note>
</div>
<p>
  <s xml:id="echoid-s2846" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s2847" xml:space="preserve">Subdo tantùm, ſi oculus uſquam intra ſpacium ZB ſtatua-<lb/>tur, viſionem indè confuſam, aut nullam evadere; </s>
  <s xml:id="echoid-s2848" xml:space="preserve">quia nempe tunc <lb/>reflexi præcipui (ſeu rectiſſimi) oculum convergentes appellent.</s>
  <s xml:id="echoid-s2849" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2850" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s2851" xml:space="preserve">Ex his porrò facilè refelluntur, quæ de imaginis loco pleni-<lb/>que tradunt omnes Optici; </s>
  <s xml:id="echoid-s2852" xml:space="preserve">cum illis noviſſimus _Honor. </s>
  <s xml:id="echoid-s2853" xml:space="preserve">Fabri;_ </s>
  <s xml:id="echoid-s2854" xml:space="preserve">juxta <lb/>quorum doctrinam imago à puncto reflectionis tanto diſtat intervallo, <lb/>quanto punctum radians ab eodem ſemovetur; </s>
  <s xml:id="echoid-s2855" xml:space="preserve">ità quidem ut Sol ex hu-<lb/>juſmodi reflectione conſpicuus ad tantam, quantam directè ſpectatus, di-<lb/>ſtantiam (eorum inſiſtendo ſententiæ) debeat apparere. </s>
  <s xml:id="echoid-s2856" xml:space="preserve">quod im-<lb/>mane quantum experientiæ refragatur. </s>
  <s xml:id="echoid-s2857" xml:space="preserve">etenim ſi Soli exponatur _Spt-_ <lb/>_culnm_ RB, (concavum, aut convexum) ſic ut ei Sol quaſi perpen-<lb/>diculariter immineat, oculúſque prope axem BC conſtituatur uſpi-<lb/>am; </s>
  <s xml:id="echoid-s2858" xml:space="preserve">ferè circa punctum Z, arbitrante ſenſu, luculenta Solis imago <lb/>ſeſe præbebit oculo conſpiciendam; </s>
  <s xml:id="echoid-s2859" xml:space="preserve">id quod juxta ratiocinium no-<lb/>ſtrum neceſſariò debuit evenire. </s>
  <s xml:id="echoid-s2860" xml:space="preserve">verùm hic error (in Opticâ capitalis, <lb/>&amp; </s>
  <s xml:id="echoid-s2861" xml:space="preserve">quo non ablegato nulla phænomeni cujuſcunque ratio veriſimilis <lb/>conſtabit) ubique ſe objiciet refutandum. </s>
  <s xml:id="echoid-s2862" xml:space="preserve">hîc itaque pluribus parco; <lb/></s>
  <s xml:id="echoid-s2863" xml:space="preserve">pergóque verſus oculum extra radiationis axem poſitum; </s>
  <s xml:id="echoid-s2864" xml:space="preserve">poſtquam <lb/>unicam hanc præcedentibus adnexam obnſervationem ſubjecero.</s>
  <s xml:id="echoid-s2865" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2866" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s2867" xml:space="preserve">Majoris Sphæræ portio vehementiùs urit; </s>
  <s xml:id="echoid-s2868" xml:space="preserve">ut &amp; </s>
  <s xml:id="echoid-s2869" xml:space="preserve">Objectum <lb/>viſibile clariùs atque diſtinctiùs repræſentat, quàm minoris æq@alem <lb/>obtinens latitudinem portio.</s>
  <s xml:id="echoid-s2870" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2871" xml:space="preserve">Super eandem nempe ſubtenſam NV inſiſtant imparium circulo-<lb/>rum ſegmenta NBV, Nbv; </s>
  <s xml:id="echoid-s2872" xml:space="preserve">quorum axis AD; </s>
  <s xml:id="echoid-s2873" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2874" xml:space="preserve">in hoc circulo-<lb/>rum centra C, c; </s>
  <s xml:id="echoid-s2875" xml:space="preserve">conſtat ut minoris peripheriam Nbv extra majoris <lb/>NBV jacere; </s>
  <s xml:id="echoid-s2876" xml:space="preserve">ità majoris centrum C infra minoris centrum c ex-<lb/>
<anchor type="note" xlink:label="note-0069-01a" xlink:href="note-0069-01"/>
iſtere. </s>
  <s xml:id="echoid-s2877" xml:space="preserve">biſecentur jam Semidiametri CB, cb in Z, z; </s>
  <s xml:id="echoid-s2878" xml:space="preserve">ducantúrque <lb/>tangentes BT, bt; </s>
  <s xml:id="echoid-s2879" xml:space="preserve">bíſque ductæ CN, cN occurrant punctis E, e; <lb/></s>
  <s xml:id="echoid-s2880" xml:space="preserve">denuò radii PN axi paralleli ſit ad peripheriam NBV reflexus NK; </s>
  <s xml:id="echoid-s2881" xml:space="preserve"><lb/>ad ipſam verò Nbv ſit ejuſdem reflexus Nk; </s>
  <s xml:id="echoid-s2882" xml:space="preserve">liquidiſſimè jam patet <lb/>quòd ſit Ne &amp;</s>
  <s xml:id="echoid-s2883" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s2884" xml:space="preserve">NE; </s>
  <s xml:id="echoid-s2885" xml:space="preserve">hoc eſt quòd dupla zk major ſit duplâ ZK; </s>
  <s xml:id="echoid-s2886" xml:space="preserve"><lb/>adeóque ſimpla zk major ſimplâ ZK. </s>
  <s xml:id="echoid-s2887" xml:space="preserve">majoris itaque Sphæræ por-<lb/>tio ſtrictiores intra terminos illabentem lucem cogit; </s>
  <s xml:id="echoid-s2888" xml:space="preserve">adeóque po-<lb/>tentiùs operatur; </s>
  <s xml:id="echoid-s2889" xml:space="preserve">eâdemque de cauſa rem objectam illuſtriù@ atque
<pb o="52" file="0070" n="70" rhead=""/>
diſtinctius exhibet obtuenti. </s>
  <s xml:id="echoid-s2890" xml:space="preserve">quod erat propoſitum oſtendere. </s>
  <s xml:id="echoid-s2891" xml:space="preserve">Et <lb/>hæc quidem ad locum imaginis determinandum attinentia pleraque <lb/>propter oculum in axe ſitum ſuffecerit attigiſſe. </s>
  <s xml:id="echoid-s2892" xml:space="preserve">Supereſt ut idem <lb/>oculi gratiâ ſecùs conſtituti pertentemus. </s>
  <s xml:id="echoid-s2893" xml:space="preserve">id operis ſequenti depu-<lb/>tamus.</s>
  <s xml:id="echoid-s2894" xml:space="preserve"/>
</p>
<div xml:id="echoid-div79" type="float" level="2" n="8">
<note position="right" xlink:label="note-0069-01" xlink:href="note-0069-01a" xml:space="preserve">Fig. 69.</note>
</div>
</div>
<div xml:id="echoid-div81" type="section" level="1" n="16">
<head xml:id="echoid-head19" xml:space="preserve"><emph style="sc">Lect.</emph> VI I.</head>
<p>
  <s xml:id="echoid-s2895" xml:space="preserve">I. </s>
  <s xml:id="echoid-s2896" xml:space="preserve">ID nunc agimus, ut ab infinito quoad ſenſum intervallo radiantis <lb/>puncti, è reflectione circularem ad peripheriam peracta oriun-<lb/>dæ imaginis, oculi reſpectu præter axem ſiti, locum exquiramus. <lb/></s>
  <s xml:id="echoid-s2897" xml:space="preserve">quocirca primùm ipſa recta linea determinanda venit, in qua locus <lb/>iſte verſatur; </s>
  <s xml:id="echoid-s2898" xml:space="preserve">tum ipſiſſimum præciſè punctum eſt deſignandum. </s>
  <s xml:id="echoid-s2899" xml:space="preserve">In <lb/>primi verò propoſiti gratiam hoc _Problema_ confici debet.</s>
  <s xml:id="echoid-s2900" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2901" xml:space="preserve">II. </s>
  <s xml:id="echoid-s2902" xml:space="preserve">Dato circulo reflectente BNP (cujus centrum C) rectâque <lb/>CB poſitione data; </s>
  <s xml:id="echoid-s2903" xml:space="preserve">deſignandus eſt huic parallelus radius, cujus re-<lb/>flexus per datum tranſeat punctum.</s>
  <s xml:id="echoid-s2904" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2905" xml:space="preserve">III. </s>
  <s xml:id="echoid-s2906" xml:space="preserve">Si datum punctum ( puta K) in ipſa CB exiſtat, facilimè pe-<lb/>
<anchor type="note" xlink:label="note-0070-01a" xlink:href="note-0070-01"/>
ragitur negotium. </s>
  <s xml:id="echoid-s2907" xml:space="preserve">nam ſi centro K, intervallo KC deſcribatur circulus, <lb/>ipſi reflectenti occurrens in N; </s>
  <s xml:id="echoid-s2908" xml:space="preserve">erit KN reſlexus ducti ad CB paralle-<lb/>li; </s>
  <s xml:id="echoid-s2909" xml:space="preserve">prout ex antedictis abunde perſpicuum eſt.</s>
  <s xml:id="echoid-s2910" xml:space="preserve"/>
</p>
<div xml:id="echoid-div81" type="float" level="2" n="1">
<note position="left" xlink:label="note-0070-01" xlink:href="note-0070-01a" xml:space="preserve">Fig. 70.</note>
</div>
<p>
  <s xml:id="echoid-s2911" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s2912" xml:space="preserve">Si datum punctum (puta jam X) in ipſa reflectentis circumfe-<lb/>rentia verſetur; </s>
  <s xml:id="echoid-s2913" xml:space="preserve">arcûs triſectione ſtatim exhauritur _Problema_. </s>
  <s xml:id="echoid-s2914" xml:space="preserve">Nam <lb/>ducatur XH ad BC parallela (quæ quidem ipſa uno modo proble-<lb/>mati ſatisfacit) &amp; </s>
  <s xml:id="echoid-s2915" xml:space="preserve">interceptus arcus XH ſecetur punctis N, P, ut <lb/>
<anchor type="note" xlink:label="note-0070-02a" xlink:href="note-0070-02"/>
ſint arcus XN, NP, PH æquales inter ſe; </s>
  <s xml:id="echoid-s2916" xml:space="preserve">connectantúrque rectæ <lb/>XN, NP. </s>
  <s xml:id="echoid-s2917" xml:space="preserve">dico factum. </s>
  <s xml:id="echoid-s2918" xml:space="preserve">etenim ducantur CN, XP; </s>
  <s xml:id="echoid-s2919" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2920" xml:space="preserve">patet an-<lb/>gulum CN X ipſi CNP æquari; </s>
  <s xml:id="echoid-s2921" xml:space="preserve">adeóque fore XN reflexum ipſius <lb/>PN; </s>
  <s xml:id="echoid-s2922" xml:space="preserve">quinetiam ang. </s>
  <s xml:id="echoid-s2923" xml:space="preserve">NP X æquatur angulo HXP; </s>
  <s xml:id="echoid-s2924" xml:space="preserve">proindéque <lb/>NP ipſi XH, hoc eſt ipſi BC, parallela eſt. </s>
  <s xml:id="echoid-s2925" xml:space="preserve">itaque factum.</s>
  <s xml:id="echoid-s2926" xml:space="preserve"/>
</p>
<div xml:id="echoid-div82" type="float" level="2" n="2">
<note position="left" xlink:label="note-0070-02" xlink:href="note-0070-02a" xml:space="preserve">Fig. 71.</note>
</div>
<pb o="53" file="0071" n="71" rhead=""/>
<p>
  <s xml:id="echoid-s2927" xml:space="preserve">V. </s>
  <s xml:id="echoid-s2928" xml:space="preserve">Verùm extra caſus hos, &amp; </s>
  <s xml:id="echoid-s2929" xml:space="preserve">particulares alios (mihi non incog-<lb/>nitos, at nunc άΠροσ{δι}ονύ?</s>
  <s xml:id="echoid-s2930" xml:space="preserve">?{ου}ς) _Problema_ magìs ſolidum eſt; </s>
  <s xml:id="echoid-s2931" xml:space="preserve">in ſummo <lb/>quippe gradu tale; </s>
  <s xml:id="echoid-s2932" xml:space="preserve">quatuórque ſubinde Solutiones admittens; </s>
  <s xml:id="echoid-s2933" xml:space="preserve">perque <lb/>lineam evolvi poteſt (ut alia pleraque, ſicutì pridem admonitum <lb/>nobis) ſibi peculiarem; </s>
  <s xml:id="echoid-s2934" xml:space="preserve">illam hoc modo quàm expeditiſſimè per <lb/>puncta deſcribendam: </s>
  <s xml:id="echoid-s2935" xml:space="preserve">Per datum punctum X protendatur indefinitè <lb/>recta GF. </s>
  <s xml:id="echoid-s2936" xml:space="preserve">ad datam CB parallela; </s>
  <s xml:id="echoid-s2937" xml:space="preserve">connectatúrque recta XC; <lb/></s>
  <s xml:id="echoid-s2938" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2939" xml:space="preserve">ſuper hanc ceu diametrum deſcribatur circulus XICI. </s>
  <s xml:id="echoid-s2940" xml:space="preserve">tum è <lb/>puncto C prodeant quotcunque rectæ circulum XIC ſecantes punctis <lb/>I, rectamque GF punctis H; </s>
  <s xml:id="echoid-s2941" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2942" xml:space="preserve">adſumantur in rectis CHI rectæ <lb/>IN æquales interceptis IH (ità ſcilicet ut puncta I rectas NH per-<lb/>
<anchor type="note" xlink:label="note-0071-01a" xlink:href="note-0071-01"/>
petuo biſecent) perque puncta quotvis ejuſmodi N traducta concipia-<lb/>tur linea; </s>
  <s xml:id="echoid-s2943" xml:space="preserve">nimirum hæc (quà certè nulla Sectio conica faciliùs de-<lb/>lineatur) problematis noſtri conſtructioni deſervit, ejúſque liquidò <lb/>naturam patefacit; </s>
  <s xml:id="echoid-s2944" xml:space="preserve">ſiquidem ejuſce cum dati circuli interſectiones <lb/>N (illæ verò ſubinde quatuor erunt, interdum tres (contactum e-<lb/>nim interſectionibus adnumero) nonnunquam Solummodò duæ; </s>
  <s xml:id="echoid-s2945" xml:space="preserve">pro-<lb/>ut datus circulus magnitudine præditus eſt aliâ ac aliâ; </s>
  <s xml:id="echoid-s2946" xml:space="preserve">quæ ſtrictim <lb/>adnoto tantùm, animum advertenti manifeſtè conſtitura) poſſibiles <lb/>quaſque Solutiones exhibebunt. </s>
  <s xml:id="echoid-s2947" xml:space="preserve">ducatur enim ab ipſo X ad ejuſmodi <lb/>quamvis interſectionem N recta XN; </s>
  <s xml:id="echoid-s2948" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2949" xml:space="preserve">per N tranſeat MP ad BC <lb/>parailela (vel ad GX) connexaque CN circulum XIC ſecet in I, <lb/>rectámque GX in H; </s>
  <s xml:id="echoid-s2950" xml:space="preserve">item jungatur XI. </s>
  <s xml:id="echoid-s2951" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2952" xml:space="preserve">quoniam è deſcriptæ <lb/>lineæ naturâ ſeu conſtructione eſt IH = IN; </s>
  <s xml:id="echoid-s2953" xml:space="preserve">angulúſque CIX, in <lb/>Semicirculo, rectus eſt; </s>
  <s xml:id="echoid-s2954" xml:space="preserve">erit XN = XH; </s>
  <s xml:id="echoid-s2955" xml:space="preserve">vel ang. </s>
  <s xml:id="echoid-s2956" xml:space="preserve">XNI = ang. <lb/></s>
  <s xml:id="echoid-s2957" xml:space="preserve">XHI. </s>
  <s xml:id="echoid-s2958" xml:space="preserve">atqui ang. </s>
  <s xml:id="echoid-s2959" xml:space="preserve">XHI alterno HN P par eſt. </s>
  <s xml:id="echoid-s2960" xml:space="preserve">quapropter anguli <lb/>XNI, HNP pares ſunt. </s>
  <s xml:id="echoid-s2961" xml:space="preserve">adeóque recta NX ipſius NP reflexus <lb/>erit. </s>
  <s xml:id="echoid-s2962" xml:space="preserve">quod oportebat fieri. </s>
  <s xml:id="echoid-s2963" xml:space="preserve">ſic, inquam, enodari poterat id Pro-<lb/>blematis. </s>
  <s xml:id="echoid-s2964" xml:space="preserve">at quoniam (ut innuebam ſuprà) _Geometrarum palato mi-_ <lb/>_nùs ſapiunt hujuſmodi Problematum inuſitatæ ſolutiones;_ </s>
  <s xml:id="echoid-s2965" xml:space="preserve">aliter id <lb/>(ſatis breviter atque perſpicuè) dabimus effectum hoc ſaltem eò <lb/>faciens Lemmaticum Problema præmittentes.</s>
  <s xml:id="echoid-s2966" xml:space="preserve"/>
</p>
<div xml:id="echoid-div83" type="float" level="2" n="3">
<note position="right" xlink:label="note-0071-01" xlink:href="note-0071-01a" xml:space="preserve">Fig. 72.</note>
</div>
<p>
  <s xml:id="echoid-s2967" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s2968" xml:space="preserve">Dato circulo (cujus poſitione data diameter GF) &amp; </s>
  <s xml:id="echoid-s2969" xml:space="preserve">puncto <lb/>C in ejuſce circumſerentia quoque dato; </s>
  <s xml:id="echoid-s2970" xml:space="preserve">per hoc recta ducatur, cujus <lb/>pars diametro circumferentiæ que interjecta æquetur datæ rectæ Z.</s>
  <s xml:id="echoid-s2971" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s2972" xml:space="preserve">Id ſic exequimur. </s>
  <s xml:id="echoid-s2973" xml:space="preserve">Connectatur recta CF; </s>
  <s xml:id="echoid-s2974" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2975" xml:space="preserve">huic perpen-<lb/>dicularis ducatur recta FV; </s>
  <s xml:id="echoid-s2976" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2977" xml:space="preserve">accipiatur ad ipſas Z, GF tertia <lb/>proportionalis P; </s>
  <s xml:id="echoid-s2978" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2979" xml:space="preserve">per G angulo CFV inſeratur recta RS par <lb/>ipſi P (id autem quomodò præſtandum, edocuimus ſupra) tum per C
<pb o="54" file="0072" n="72" rhead=""/>
ducatur CHL ad RS parallela; </s>
  <s xml:id="echoid-s2980" xml:space="preserve">erit intercepta HL (quod requiri-<lb/>tur) æqualis ipſi Z. </s>
  <s xml:id="echoid-s2981" xml:space="preserve">Nam connectatur CG; </s>
  <s xml:id="echoid-s2982" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s2983" xml:space="preserve">huic perpendicu-<lb/>laris ducatur GT; </s>
  <s xml:id="echoid-s2984" xml:space="preserve">ad CF proinde parallela. </s>
  <s xml:id="echoid-s2985" xml:space="preserve">quia jam ang. </s>
  <s xml:id="echoid-s2986" xml:space="preserve">GCT <lb/> = CGR = FSR, liquet rectangula trigona CGT, RFS aſſi-<lb/>milari. </s>
  <s xml:id="echoid-s2987" xml:space="preserve">adeóque fore CT. </s>
  <s xml:id="echoid-s2988" xml:space="preserve">CG :</s>
  <s xml:id="echoid-s2989" xml:space="preserve">: SR . </s>
  <s xml:id="echoid-s2990" xml:space="preserve">SF. </s>
  <s xml:id="echoid-s2991" xml:space="preserve">item (ob ſimilitudinem <lb/>triangulorum CGH, SFG) eſt CG. </s>
  <s xml:id="echoid-s2992" xml:space="preserve">GH :</s>
  <s xml:id="echoid-s2993" xml:space="preserve">: SF. </s>
  <s xml:id="echoid-s2994" xml:space="preserve">FG. </s>
  <s xml:id="echoid-s2995" xml:space="preserve">erit igi-<lb/>tur ex æquo CT. </s>
  <s xml:id="echoid-s2996" xml:space="preserve">GH :</s>
  <s xml:id="echoid-s2997" xml:space="preserve">: SR. </s>
  <s xml:id="echoid-s2998" xml:space="preserve">FG. </s>
  <s xml:id="echoid-s2999" xml:space="preserve">(hoc eſt) :</s>
  <s xml:id="echoid-s3000" xml:space="preserve">: FG. </s>
  <s xml:id="echoid-s3001" xml:space="preserve">Z. </s>
  <s xml:id="echoid-s3002" xml:space="preserve">verùm <lb/>eſt CT. </s>
  <s xml:id="echoid-s3003" xml:space="preserve">FG :</s>
  <s xml:id="echoid-s3004" xml:space="preserve">: CH. </s>
  <s xml:id="echoid-s3005" xml:space="preserve">FH :</s>
  <s xml:id="echoid-s3006" xml:space="preserve">: HG. </s>
  <s xml:id="echoid-s3007" xml:space="preserve">HL. </s>
  <s xml:id="echoid-s3008" xml:space="preserve">permutandóque CT. </s>
  <s xml:id="echoid-s3009" xml:space="preserve">HG <lb/>:</s>
  <s xml:id="echoid-s3010" xml:space="preserve">: FG. </s>
  <s xml:id="echoid-s3011" xml:space="preserve">HL. </s>
  <s xml:id="echoid-s3012" xml:space="preserve">quare FG. </s>
  <s xml:id="echoid-s3013" xml:space="preserve">Z :</s>
  <s xml:id="echoid-s3014" xml:space="preserve">: FG. </s>
  <s xml:id="echoid-s3015" xml:space="preserve">HL. </s>
  <s xml:id="echoid-s3016" xml:space="preserve">liquet igitur HL ipſi Z <lb/>datæ æquari: </s>
  <s xml:id="echoid-s3017" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s3018" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3019" xml:space="preserve">F.</s>
  <s xml:id="echoid-s3020" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3021" xml:space="preserve">Plures eſſe caſus poſſunt; </s>
  <s xml:id="echoid-s3022" xml:space="preserve">ut nempe punctum L ſit intra Semicircu-<lb/>lum GCF (ídque poſitum inter puncta C, G, vel inter ipſa C, F) vel <lb/>
<anchor type="note" xlink:label="note-0072-01a" xlink:href="note-0072-01"/>
in altero Semicirculo GE F, ultra GF ſito reſpectu puncti C; </s>
  <s xml:id="echoid-s3023" xml:space="preserve">ſed <lb/>hæc una conſtructio ſimul ac demonſtratio pariter omnibus convenit; <lb/></s>
  <s xml:id="echoid-s3024" xml:space="preserve">ut pluribus huc non ſit opus.</s>
  <s xml:id="echoid-s3025" xml:space="preserve"/>
</p>
<div xml:id="echoid-div84" type="float" level="2" n="4">
<note position="left" xlink:label="note-0072-01" xlink:href="note-0072-01a" xml:space="preserve">Fig. 73, 74.</note>
</div>
<p>
  <s xml:id="echoid-s3026" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s3027" xml:space="preserve">Adnotetur ſaltem quoad iſtos caſus, quod ſicuti per punctum <lb/>G (ut antea commoſtratum) aliquando quatnor rectæ duci poſſunt <lb/>datam adæquantes, rectíſque FC, FV terminatæ; </s>
  <s xml:id="echoid-s3028" xml:space="preserve">binæ ſcilicet inter <lb/>angulum quo punctum G continetur, alteræque totidem extra ipſum; <lb/></s>
  <s xml:id="echoid-s3029" xml:space="preserve">nonnunquam verò tres ſolæ; </s>
  <s xml:id="echoid-s3030" xml:space="preserve">quum data recta minima continget eſſe <lb/>cunctarum, quæ dicto punctum G continenti angulo poſſunt interſeri; </s>
  <s xml:id="echoid-s3031" xml:space="preserve"><lb/>ſubinde tantùm duæ, quando data tali minimæ cedit; </s>
  <s xml:id="echoid-s3032" xml:space="preserve">ita reſpectivè <lb/>Problema jam expoſitum plures totidem ſolutiones accipit. </s>
  <s xml:id="echoid-s3033" xml:space="preserve">Sanè <lb/>quò major eſt hîc data Z, cò minor evadet intercepta RS; </s>
  <s xml:id="echoid-s3034" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3035" xml:space="preserve">viciſſim <lb/>quò minor RS, eò major ipſa IZ; </s>
  <s xml:id="echoid-s3036" xml:space="preserve">unde ſi fuerit RS omnium mini-<lb/>ma, quæ angulo CFV punctum G capienti inſeri poſſunt, etiam HL <lb/>maxima erit è C prodeuntium rectarum, quæ inter diametrum GF, <lb/>&amp; </s>
  <s xml:id="echoid-s3037" xml:space="preserve">Semicirculum GEF comprchendi poſſunt. </s>
  <s xml:id="echoid-s3038" xml:space="preserve">unde Poriſmatis loco patet, <lb/>è ſupradictis, quo pacto talis maxima ducí poſſit; </s>
  <s xml:id="echoid-s3039" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3040" xml:space="preserve">hoc ipſum Pro-<lb/>blema penitus determinari. </s>
  <s xml:id="echoid-s3041" xml:space="preserve">quod attendenti non obſcurum innuiſſe <lb/>ſatìs videtur. </s>
  <s xml:id="echoid-s3042" xml:space="preserve">jam ad principalis quæſiti rcſolutionem accedimus; </s>
  <s xml:id="echoid-s3043" xml:space="preserve">ità <lb/>jam brevitur propoſiti.</s>
  <s xml:id="echoid-s3044" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3045" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s3046" xml:space="preserve">Per datum punctum X rectam ducere, cujus reflexus datæ <lb/>
<anchor type="note" xlink:label="note-0072-02a" xlink:href="note-0072-02"/>
poſitione rectæ BC ſit parallelus.</s>
  <s xml:id="echoid-s3047" xml:space="preserve"/>
</p>
<div xml:id="echoid-div85" type="float" level="2" n="5">
<note position="left" xlink:label="note-0072-02" xlink:href="note-0072-02a" xml:space="preserve">Fig. 73, 74.</note>
</div>
<p>
  <s xml:id="echoid-s3048" xml:space="preserve">Id ſic efficitur. </s>
  <s xml:id="echoid-s3049" xml:space="preserve">Centro X per C deſcribæur circulus GLFC; <lb/></s>
  <s xml:id="echoid-s3050" xml:space="preserve">item per X ducatur GF ad BC parallela; </s>
  <s xml:id="echoid-s3051" xml:space="preserve">tum ex C prjoiciatur <lb/>recta, cujus ſecundum Lemma mox præcedens, intercepta pars H L <lb/>æquetur Semidiametro reflectentis circuli; </s>
  <s xml:id="echoid-s3052" xml:space="preserve">quæ &amp; </s>
  <s xml:id="echoid-s3053" xml:space="preserve">illum ſecet in N;</s>
  <s xml:id="echoid-s3054" xml:space="preserve">
<pb o="55" file="0073" n="73" rhead=""/>
ductæ XN reflexus (puta NP) ipſi BC parallelus erit. </s>
  <s xml:id="echoid-s3055" xml:space="preserve">Nam con-<lb/>nexis XC, XL; </s>
  <s xml:id="echoid-s3056" xml:space="preserve">quoniam CN = HL, &amp; </s>
  <s xml:id="echoid-s3057" xml:space="preserve">CX = LX; </s>
  <s xml:id="echoid-s3058" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3059" xml:space="preserve">anguli <lb/>XCL, XLC pares ſunt; </s>
  <s xml:id="echoid-s3060" xml:space="preserve">erit XH = XN. </s>
  <s xml:id="echoid-s3061" xml:space="preserve">quapropter erit NP <lb/>ad XH, vel BC parallelus: </s>
  <s xml:id="echoid-s3062" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s3063" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3064" xml:space="preserve">F.</s>
  <s xml:id="echoid-s3065" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3066" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s3067" xml:space="preserve">Ex hac conſtructione, cum præmiſſi lemmatis ſolutione colla-<lb/>tâ diluceſcet hujuſmodi non ultra quatuor reflexos per idem quodcun-<lb/>que punctum, ceu X, tranſire; </s>
  <s xml:id="echoid-s3068" xml:space="preserve">quorum duo ad unas axis partes inci-<lb/>dentibus, reliqui ad alteras conveniunt. </s>
  <s xml:id="echoid-s3069" xml:space="preserve">adparebit ctiam ſi CN <lb/>major ſit, quam ut ci par HL rectâ GF, Semicirculóque GEF <lb/>intercipi poſſit; </s>
  <s xml:id="echoid-s3070" xml:space="preserve">quòd ad axis partes, ad quas ipſum X ponitur, om-<lb/>
<anchor type="note" xlink:label="note-0073-01a" xlink:href="note-0073-01"/>
nino nullus per hoc punctum reflexus meabit; </s>
  <s xml:id="echoid-s3071" xml:space="preserve">quinetiam ſi CN <lb/>tanta ſit, ut ci par una tantùm ejuſmodi recta poſſit intercipi, quòd <lb/>unicus per ipſum X reflexus iter ſuſcipiet. </s>
  <s xml:id="echoid-s3072" xml:space="preserve">tales, inquam, expoſiti <lb/>problematis determinationes hanc conſtructionem haud obſcurè ſe-<lb/>quuntur; </s>
  <s xml:id="echoid-s3073" xml:space="preserve">quas certè tu meliùs uno mentis (haud dormitantis) ictu <lb/>perſpexeris, quàm ego pluribus verbis explicâro.</s>
  <s xml:id="echoid-s3074" xml:space="preserve"/>
</p>
<div xml:id="echoid-div86" type="float" level="2" n="6">
<note position="right" xlink:label="note-0073-01" xlink:href="note-0073-01a" xml:space="preserve">Fig. 75.</note>
</div>
<p>
  <s xml:id="echoid-s3075" xml:space="preserve">X. </s>
  <s xml:id="echoid-s3076" xml:space="preserve">Exhinc itaque denuò rectam (ſeurectas) ſatìs definivimus, in <lb/>qua (vel ìn quibus) puncti radiantis lmago, reſpectu visûs utcunque <lb/>poſitione datum centrum habentis, conſiſtit. </s>
  <s xml:id="echoid-s3077" xml:space="preserve">ad ejus jam præciſiorem <lb/>locum inveſtigandum accingemur; </s>
  <s xml:id="echoid-s3078" xml:space="preserve">in iſtarum rectâ quâpiam exiſten-<lb/>tem.</s>
  <s xml:id="echoid-s3079" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3080" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s3081" xml:space="preserve">Huc adnotetur imprimìs, quòd ſi duorum ad eaſdem axis par-<lb/>tes incidentium parallelorum (NP, RS) reflexi ſint N Π, R σ; <lb/></s>
  <s xml:id="echoid-s3082" xml:space="preserve">erit arcus NR, vel PS arcûs Π σ ſubtriplus. </s>
  <s xml:id="echoid-s3083" xml:space="preserve">Concurrant enim dicti <lb/>reflexi in X; </s>
  <s xml:id="echoid-s3084" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3085" xml:space="preserve">Connectatur recta R Π. </s>
  <s xml:id="echoid-s3086" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3087" xml:space="preserve">quoniam, è præmonitis, <lb/>
<anchor type="note" xlink:label="note-0073-02a" xlink:href="note-0073-02"/>
angulus NX R duplus eſt anguli arcui NR ad centrum inſiſtentis; <lb/></s>
  <s xml:id="echoid-s3088" xml:space="preserve">crit idem angulus NXR anguli N Π R quadruplus. </s>
  <s xml:id="echoid-s3089" xml:space="preserve">quapropter erit <lb/>ang. </s>
  <s xml:id="echoid-s3090" xml:space="preserve">NXR - ang. </s>
  <s xml:id="echoid-s3091" xml:space="preserve">N Π R triplus anguli N Π R, hoc eſt angulus <lb/>XR Π anguli N Π R triplus. </s>
  <s xml:id="echoid-s3092" xml:space="preserve">unde quoque triplus erit arcus Π σ ipſius <lb/>NR : </s>
  <s xml:id="echoid-s3093" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s3094" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3095" xml:space="preserve">D.</s>
  <s xml:id="echoid-s3096" xml:space="preserve"/>
</p>
<div xml:id="echoid-div87" type="float" level="2" n="7">
<note position="right" xlink:label="note-0073-02" xlink:href="note-0073-02a" xml:space="preserve">Fig. 76.</note>
</div>
<p>
  <s xml:id="echoid-s3097" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s3098" xml:space="preserve">Iiſdem ſtantibus dico fore RX (obliquioris reflexi partem <lb/>incidentiæ concursûſque punctis interceptam) majorem quadrante to-<lb/>tius reflexi R σ. </s>
  <s xml:id="echoid-s3099" xml:space="preserve">Nam, ductis ſubtenſis NR, Π σ; </s>
  <s xml:id="echoid-s3100" xml:space="preserve">erit I. </s>
  <s xml:id="echoid-s3101" xml:space="preserve">3 :</s>
  <s xml:id="echoid-s3102" xml:space="preserve">: <lb/>arc. </s>
  <s xml:id="echoid-s3103" xml:space="preserve">NR. </s>
  <s xml:id="echoid-s3104" xml:space="preserve">Π σ. </s>
  <s xml:id="echoid-s3105" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s3106" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3107" xml:space="preserve">recta NR. </s>
  <s xml:id="echoid-s3108" xml:space="preserve">Π σ :</s>
  <s xml:id="echoid-s3109" xml:space="preserve">: RX. </s>
  <s xml:id="echoid-s3110" xml:space="preserve">X Π &amp;</s>
  <s xml:id="echoid-s3111" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3112" xml:space="preserve">RX. </s>
  <s xml:id="echoid-s3113" xml:space="preserve">X σ (quia <lb/>ſcilicet eſt X Π &amp;</s>
  <s xml:id="echoid-s3114" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3115" xml:space="preserve">X σ). </s>
  <s xml:id="echoid-s3116" xml:space="preserve">igitur eſt X σ minor triplâ RX; </s>
  <s xml:id="echoid-s3117" xml:space="preserve">compo-<lb/>nendóque minor erit R σ quadruplâ RX: </s>
  <s xml:id="echoid-s3118" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s3119" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3120" xml:space="preserve">D.</s>
  <s xml:id="echoid-s3121" xml:space="preserve"/>
</p>
<pb o="56" file="0074" n="74" rhead=""/>
<p>
  <s xml:id="echoid-s3122" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s3123" xml:space="preserve">Item, dico fore NX (rectioris itidem reflexi concursûs <lb/>incidentiæque punctis interjectam partem) minorem quartâ parte <lb/>totius N Π. </s>
  <s xml:id="echoid-s3124" xml:space="preserve">Etenim fiat ang. </s>
  <s xml:id="echoid-s3125" xml:space="preserve">HR Π = ang. </s>
  <s xml:id="echoid-s3126" xml:space="preserve">N Π R; </s>
  <s xml:id="echoid-s3127" xml:space="preserve">quapropter <lb/>erit HR = H Π; </s>
  <s xml:id="echoid-s3128" xml:space="preserve">adeóque 2 H Π = HR + H Π &amp;</s>
  <s xml:id="echoid-s3129" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3130" xml:space="preserve">R Π &amp;</s>
  <s xml:id="echoid-s3131" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3132" xml:space="preserve">N Π. <lb/></s>
  <s xml:id="echoid-s3133" xml:space="preserve">item quoniam ang. </s>
  <s xml:id="echoid-s3134" xml:space="preserve">RHN = 2 ang. </s>
  <s xml:id="echoid-s3135" xml:space="preserve">HR Π = ang. </s>
  <s xml:id="echoid-s3136" xml:space="preserve">XRH; </s>
  <s xml:id="echoid-s3137" xml:space="preserve">eſt <lb/>XH = XR &amp;</s>
  <s xml:id="echoid-s3138" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3139" xml:space="preserve">XN. </s>
  <s xml:id="echoid-s3140" xml:space="preserve">quum itaque ſit H Π major ſemiſſe totius N Π; </s>
  <s xml:id="echoid-s3141" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s3142" xml:space="preserve">XH major ſemiſſe reſidui NH; </s>
  <s xml:id="echoid-s3143" xml:space="preserve">liquet totam X Π majorem eſſe <lb/>triplâ X N; </s>
  <s xml:id="echoid-s3144" xml:space="preserve">ſeu totam N Π majorem eſſe quadruplâ N X: </s>
  <s xml:id="echoid-s3145" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s3146" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3147" xml:space="preserve">D.</s>
  <s xml:id="echoid-s3148" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3149" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s3150" xml:space="preserve">Hinc perſpicuum eſt, ſi fuerit NZ reflexi N Π quadrans, <lb/>quòd nullus alter hujuſmodi reflexus punctum Z permeabit. </s>
  <s xml:id="echoid-s3151" xml:space="preserve">Etenim <lb/>alterius cujuſvis reflexus permeare dicatur; </s>
  <s xml:id="echoid-s3152" xml:space="preserve">erit igitur, ſi obliquior <lb/>is fuerit, NZ &amp;</s>
  <s xml:id="echoid-s3153" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3154" xml:space="preserve">{1/4} N Π; </s>
  <s xml:id="echoid-s3155" xml:space="preserve">ſin rectior fuerit, erit NZ &amp;</s>
  <s xml:id="echoid-s3156" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3157" xml:space="preserve">{1/4} N Π (ni-<lb/>mirum è proximè demonſtratis hæc conſequuntur) quæ repugnant <lb/>hypotheſi.</s>
  <s xml:id="echoid-s3158" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3159" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s3160" xml:space="preserve">Quinetiam ipſi N Π propiùs adjacentium occurſus puncto Z vi-<lb/>
<anchor type="note" xlink:label="note-0074-01a" xlink:href="note-0074-01"/>
ciniores ſunt, hinc indè. </s>
  <s xml:id="echoid-s3161" xml:space="preserve">Secent. </s>
  <s xml:id="echoid-s3162" xml:space="preserve">inquam, radiorum LM, RS reflexi <lb/>L μ, R σ ipſam N Π punctis Y, X; </s>
  <s xml:id="echoid-s3163" xml:space="preserve">iſtæ quidem (rectior) in Y, hic <lb/>(obliquior) in X; </s>
  <s xml:id="echoid-s3164" xml:space="preserve">erit ZY &amp;</s>
  <s xml:id="echoid-s3165" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3166" xml:space="preserve">ZX. </s>
  <s xml:id="echoid-s3167" xml:space="preserve">Nam connectantur R Π, <lb/>L Π; </s>
  <s xml:id="echoid-s3168" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3169" xml:space="preserve">fiat ang. </s>
  <s xml:id="echoid-s3170" xml:space="preserve">Π LH = ang. </s>
  <s xml:id="echoid-s3171" xml:space="preserve">N Π L; </s>
  <s xml:id="echoid-s3172" xml:space="preserve">ducantúrque rectæ RH, <lb/>RY. </s>
  <s xml:id="echoid-s3173" xml:space="preserve">eſtque RH &amp;</s>
  <s xml:id="echoid-s3174" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3175" xml:space="preserve">LH = H Π; </s>
  <s xml:id="echoid-s3176" xml:space="preserve">adeóque ang. </s>
  <s xml:id="echoid-s3177" xml:space="preserve">H Π; </s>
  <s xml:id="echoid-s3178" xml:space="preserve">R &amp;</s>
  <s xml:id="echoid-s3179" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3180" xml:space="preserve">ang. <lb/></s>
  <s xml:id="echoid-s3181" xml:space="preserve">HR Π; </s>
  <s xml:id="echoid-s3182" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3183" xml:space="preserve">proinde ang. </s>
  <s xml:id="echoid-s3184" xml:space="preserve">NHR &amp;</s>
  <s xml:id="echoid-s3185" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3186" xml:space="preserve">2 ang. </s>
  <s xml:id="echoid-s3187" xml:space="preserve">H Π R. </s>
  <s xml:id="echoid-s3188" xml:space="preserve">item YR &amp;</s>
  <s xml:id="echoid-s3189" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3190" xml:space="preserve">YL <lb/> = YH; </s>
  <s xml:id="echoid-s3191" xml:space="preserve">proindéque rurſus ang. </s>
  <s xml:id="echoid-s3192" xml:space="preserve">NYR &amp;</s>
  <s xml:id="echoid-s3193" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3194" xml:space="preserve">2 ang. </s>
  <s xml:id="echoid-s3195" xml:space="preserve">YHR. </s>
  <s xml:id="echoid-s3196" xml:space="preserve">quare <lb/>multo minor eſt ang. </s>
  <s xml:id="echoid-s3197" xml:space="preserve">NY R quadruplo N Π R eſt autem ang. </s>
  <s xml:id="echoid-s3198" xml:space="preserve">NXR <lb/>quadruplus anguli N Π R; </s>
  <s xml:id="echoid-s3199" xml:space="preserve">igitur ang. </s>
  <s xml:id="echoid-s3200" xml:space="preserve">NXR &amp;</s>
  <s xml:id="echoid-s3201" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3202" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s3203" xml:space="preserve">NYR. </s>
  <s xml:id="echoid-s3204" xml:space="preserve">pona-<lb/>tur jam, ſi fieri poteſt, punctum X ipſis Y, Z interjacere. </s>
  <s xml:id="echoid-s3205" xml:space="preserve">erit igitur <lb/>angulus externus NYR interno NXR major; </s>
  <s xml:id="echoid-s3206" xml:space="preserve">atqui minor oſtenſus <lb/>eſt. </s>
  <s xml:id="echoid-s3207" xml:space="preserve">quæ repugnant. </s>
  <s xml:id="echoid-s3208" xml:space="preserve">itaque potiùs eſt ZY &amp;</s>
  <s xml:id="echoid-s3209" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3210" xml:space="preserve">ZX: </s>
  <s xml:id="echoid-s3211" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s3212" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3213" xml:space="preserve">D.</s>
  <s xml:id="echoid-s3214" xml:space="preserve"/>
</p>
<div xml:id="echoid-div88" type="float" level="2" n="8">
<note position="left" xlink:label="note-0074-01" xlink:href="note-0074-01a" xml:space="preserve">Fig. 77.</note>
</div>
<p>
  <s xml:id="echoid-s3215" xml:space="preserve">Ad alteras partes haud abſimilis erit diſcurſus; </s>
  <s xml:id="echoid-s3216" xml:space="preserve">parco faſtidioſæ re-<lb/>petitioni. </s>
  <s xml:id="echoid-s3217" xml:space="preserve">‖</s>
</p>
<p>
  <s xml:id="echoid-s3218" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s3219" xml:space="preserve">Hinc obiter patet ad eaſdem partes incidentium reflexos ſeſe <lb/>priùs (velut ad φ) quàm ipſum N Π decuſſare.</s>
  <s xml:id="echoid-s3220" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3221" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s3222" xml:space="preserve">Quinimò rurſus hinc conſtat ad eaſdem axis partes plures <lb/>duobus in uno puncto reflexos non concurrere.</s>
  <s xml:id="echoid-s3223" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3224" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s3225" xml:space="preserve">Demùm (utaliquando tandem deſtinatum attingamus ſcopum)
<pb o="57" file="0075" n="75" rhead=""/>
è dictis colligatur licet, quòd oculo, cujus Centrum O uſpiam in <lb/>ipſa N Π ponitur, circa punctum Z (ipſam N Π prænotato modo <lb/>quadriſecans) radiantis imago conſpicietur. </s>
  <s xml:id="echoid-s3226" xml:space="preserve">Sit enim pupillæ (prout <lb/>antehac aliquoties) diameter EF; </s>
  <s xml:id="echoid-s3227" xml:space="preserve">per eujuſce terminos tranſeant <lb/>radiorum LM, RS reflexi LE, RF; </s>
  <s xml:id="echoid-s3228" xml:space="preserve">quorum iſte ſecet ipſum N Π in <lb/>Y, hic in X. </s>
  <s xml:id="echoid-s3229" xml:space="preserve">quoniam igitur radiorum obliquiorum ipſo RS, <lb/>rectiorum ipſo LM nullus oculum intrabit; </s>
  <s xml:id="echoid-s3230" xml:space="preserve">utì ſuprà non ſemel argu-<lb/>mentati ſumus, intra ſpatium XY neceſſario conſiſtet imago. </s>
  <s xml:id="echoid-s3231" xml:space="preserve">quineti-<lb/>
<anchor type="note" xlink:label="note-0075-01a" xlink:href="note-0075-01"/>
am cùm radiorum arcui LR incidentium qui prope punctum Z re-<lb/>flectuntur axi N Π propiùs adjacentes perpendicularius oculum fe-<lb/>riunt, ìdque ſpiſſiùs (ut ex analogia par eſt exiſtimare; </s>
  <s xml:id="echoid-s3232" xml:space="preserve">nec enim <lb/>id operoſiùs aggrediar demonſtrare) propter aliquoties expoſitas <lb/>cauſas ab eo videbuntur obtutum afficientes radii promanare; </s>
  <s xml:id="echoid-s3233" xml:space="preserve">hoc eſt <lb/>ad ipſum imago conſiſtet. </s>
  <s xml:id="echoid-s3234" xml:space="preserve">Accedit quod ob anguſtiam pupillæ ſpa-<lb/>tium XY ſatìs modicum exiſtit; </s>
  <s xml:id="echoid-s3235" xml:space="preserve">ut puncti modum vix excedere vi-<lb/>deatur.</s>
  <s xml:id="echoid-s3236" xml:space="preserve"/>
</p>
<div xml:id="echoid-div89" type="float" level="2" n="9">
<note position="right" xlink:label="note-0075-01" xlink:href="note-0075-01a" xml:space="preserve">Fig. 78.</note>
</div>
<p>
  <s xml:id="echoid-s3237" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s3238" xml:space="preserve">Subdo; </s>
  <s xml:id="echoid-s3239" xml:space="preserve">ſi ſtatuatur oculi centrum uſpiam in ZN; </s>
  <s xml:id="echoid-s3240" xml:space="preserve">iſque ver-<lb/>ſus partes N obvertatur; </s>
  <s xml:id="echoid-s3241" xml:space="preserve">objectum confuſiùs apparere; </s>
  <s xml:id="echoid-s3242" xml:space="preserve">quippe cùm <lb/>reflexi viſum convergentes appellant; </s>
  <s xml:id="echoid-s3243" xml:space="preserve">vel quoniam imago Z tunc pone <lb/>viſum conſiſtit.</s>
  <s xml:id="echoid-s3244" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3245" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s3246" xml:space="preserve">Hinc à ſpeculo Cavo tantùm una repræſentatur Imago, ſaltem <lb/>bene diſtincta. </s>
  <s xml:id="echoid-s3247" xml:space="preserve">Nam in duorum reflexorum N Π;</s>
  <s xml:id="echoid-s3248" xml:space="preserve">, R σ concurſu X <lb/>ſtatuatur oculi centrum; </s>
  <s xml:id="echoid-s3249" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3250" xml:space="preserve">ſit R ζ = 1/4 R σ; </s>
  <s xml:id="echoid-s3251" xml:space="preserve">unde R ζ &amp;</s>
  <s xml:id="echoid-s3252" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3253" xml:space="preserve">RX. <lb/></s>
  <s xml:id="echoid-s3254" xml:space="preserve">itaque ſpectabitur quæ ad ζ imago ab oculo in X collocato, verſúſque <lb/>partes NR obverſo; </s>
  <s xml:id="echoid-s3255" xml:space="preserve">ſed tum imago Z poſt oculum conſiſtit.</s>
  <s xml:id="echoid-s3256" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3257" xml:space="preserve">XXI. </s>
  <s xml:id="echoid-s3258" xml:space="preserve">Et hæc quidem rectè percepta, ſerióquè perpenſa vix addubi-<lb/>to quin facilè ſibi fidem conciliatura ſint; </s>
  <s xml:id="echoid-s3259" xml:space="preserve">nihil ut ſit opus adverſantia <lb/>_ſeu veterum Opticorum decreta, ſeu recentiorum Commenta pluribus_ <lb/>_convellere;_ </s>
  <s xml:id="echoid-s3260" xml:space="preserve">quæ certè cùm nullâ perſpicuâ ratione nituntur, tum ab <lb/>experientia plerumque diſcordant. </s>
  <s xml:id="echoid-s3261" xml:space="preserve">Cætera verò ſiqua reſtant ad hoc <lb/>argumentum ſpectantia ſtudio veſtro commendabimus elicienda; </s>
  <s xml:id="echoid-s3262" xml:space="preserve">mox <lb/>ad è ſenſibiliter finita diſtantia radiantis puncti _Symptomata_ ſimiliter <lb/>exploranda animum adjecturi.</s>
  <s xml:id="echoid-s3263" xml:space="preserve"/>
</p>
<pb o="58" file="0076" n="76"/>
</div>
<div xml:id="echoid-div91" type="section" level="1" n="17">
<head xml:id="echoid-head20" xml:space="preserve"><emph style="sc">Lect</emph>. VIII.</head>
<p>
  <s xml:id="echoid-s3264" xml:space="preserve">I QUæ radiis obveniunt à longinquo puncto manantibus, adc-<lb/>óque quaſi parallelis, ex reflectione peripheriam ad circula-<lb/>rem peractâ; </s>
  <s xml:id="echoid-s3265" xml:space="preserve">ubinam &amp; </s>
  <s xml:id="echoid-s3266" xml:space="preserve">quouſque vel ſibimet ipſis occurrunt, vel ax-<lb/>em interſecant; </s>
  <s xml:id="echoid-s3267" xml:space="preserve">quo loco radians oculo ubicunque conſtituto repræ-<lb/>ſentant, in poſtremis eſt diſſertatum. </s>
  <s xml:id="echoid-s3268" xml:space="preserve">ad punctum jam accedimus ra-<lb/>dios ejiciens ſenſibiliter divergentes. </s>
  <s xml:id="echoid-s3269" xml:space="preserve">Et hujuſmodi quidem puncto, <lb/>quanquam ſeu in obverſas circuli Convexas partes ſeu ad concavas <lb/>radiet communia pleraque ſymptomata conveniunt; </s>
  <s xml:id="echoid-s3270" xml:space="preserve">tamen communi <lb/>fretus _Opticorum_ exemplo, præſertímque majoris evidentiæ causâ, <lb/>caſus iſtos diſtinctè proſequemur; </s>
  <s xml:id="echoid-s3271" xml:space="preserve">illum fuſiùs imprimis, hunc aliquan-<lb/>to conciſiùs. </s>
  <s xml:id="echoid-s3272" xml:space="preserve">ad rem.</s>
  <s xml:id="echoid-s3273" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3274" xml:space="preserve">II. </s>
  <s xml:id="echoid-s3275" xml:space="preserve">In circuli BNP (cujus centrum C) convexum à puncto A <lb/>quilibet incidat radius AN, íſque reflectatur in NG; </s>
  <s xml:id="echoid-s3276" xml:space="preserve">patet reflex-<lb/>um GN productum axi AC occurſurum. </s>
  <s xml:id="echoid-s3277" xml:space="preserve">nam ductâ CNE patet <lb/>GN productum angulum ANC ſecare; </s>
  <s xml:id="echoid-s3278" xml:space="preserve">nec non ideò trianguli <lb/>ANC baſin AC; </s>
  <s xml:id="echoid-s3279" xml:space="preserve">puta in K; </s>
  <s xml:id="echoid-s3280" xml:space="preserve">quo poſito.</s>
  <s xml:id="echoid-s3281" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3282" xml:space="preserve">III. </s>
  <s xml:id="echoid-s3283" xml:space="preserve">Dico fore AC. </s>
  <s xml:id="echoid-s3284" xml:space="preserve">AN :</s>
  <s xml:id="echoid-s3285" xml:space="preserve">: KC. </s>
  <s xml:id="echoid-s3286" xml:space="preserve">KN. </s>
  <s xml:id="echoid-s3287" xml:space="preserve">Nam ducatur KH ad <lb/>CN parallela. </s>
  <s xml:id="echoid-s3288" xml:space="preserve">eſt igitnr ang. </s>
  <s xml:id="echoid-s3289" xml:space="preserve">KHN = CNP = CNK = NKH. <lb/></s>
  <s xml:id="echoid-s3290" xml:space="preserve">hoc etiam è ſuperiùs generatim oftenſis conſectatur. </s>
  <s xml:id="echoid-s3291" xml:space="preserve">adeóque NH <lb/> = NK. </s>
  <s xml:id="echoid-s3292" xml:space="preserve">itâque cùm ſit AC. </s>
  <s xml:id="echoid-s3293" xml:space="preserve">AN :</s>
  <s xml:id="echoid-s3294" xml:space="preserve">: KC. </s>
  <s xml:id="echoid-s3295" xml:space="preserve">HN. </s>
  <s xml:id="echoid-s3296" xml:space="preserve">erit etiam AC. </s>
  <s xml:id="echoid-s3297" xml:space="preserve"><lb/>AN :</s>
  <s xml:id="echoid-s3298" xml:space="preserve">: KC. </s>
  <s xml:id="echoid-s3299" xml:space="preserve">KN.</s>
  <s xml:id="echoid-s3300" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3301" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s3302" xml:space="preserve">Corollarii loco notetur (ductâ CP) fore NH = NK; <lb/></s>
  <s xml:id="echoid-s3303" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3304" xml:space="preserve">triangula HNK, NCP aſſimilari; </s>
  <s xml:id="echoid-s3305" xml:space="preserve">vel eſſe HK. </s>
  <s xml:id="echoid-s3306" xml:space="preserve">HN :</s>
  <s xml:id="echoid-s3307" xml:space="preserve">: NP. </s>
  <s xml:id="echoid-s3308" xml:space="preserve"><lb/>CN.</s>
  <s xml:id="echoid-s3309" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3310" xml:space="preserve">V. </s>
  <s xml:id="echoid-s3311" xml:space="preserve">Porrò, conftantibus iiſdem, dico fore AC. </s>
  <s xml:id="echoid-s3312" xml:space="preserve">KC :</s>
  <s xml:id="echoid-s3313" xml:space="preserve">: ACq <lb/>- ANq . </s>
  <s xml:id="echoid-s3314" xml:space="preserve">CNq. </s>
  <s xml:id="echoid-s3315" xml:space="preserve">Nam eſt NP. </s>
  <s xml:id="echoid-s3316" xml:space="preserve">CN + AN. </s>
  <s xml:id="echoid-s3317" xml:space="preserve">CN = HK. <lb/></s>
  <s xml:id="echoid-s3318" xml:space="preserve">HN + AN. </s>
  <s xml:id="echoid-s3319" xml:space="preserve">CN = HK x AN. </s>
  <s xml:id="echoid-s3320" xml:space="preserve">HN x CN = AN. </s>
  <s xml:id="echoid-s3321" xml:space="preserve">HN
<pb o="59" file="0077" n="77" rhead=""/>
+ HK. </s>
  <s xml:id="echoid-s3322" xml:space="preserve">CN. </s>
  <s xml:id="echoid-s3323" xml:space="preserve">= AC. </s>
  <s xml:id="echoid-s3324" xml:space="preserve">KC + AK. </s>
  <s xml:id="echoid-s3325" xml:space="preserve">AC = AK. </s>
  <s xml:id="echoid-s3326" xml:space="preserve">KC. </s>
  <s xml:id="echoid-s3327" xml:space="preserve">verùm eſt <lb/>NP. </s>
  <s xml:id="echoid-s3328" xml:space="preserve">CN + AN. </s>
  <s xml:id="echoid-s3329" xml:space="preserve">CN = NP x AN. </s>
  <s xml:id="echoid-s3330" xml:space="preserve">CNq. </s>
  <s xml:id="echoid-s3331" xml:space="preserve">ergò erit AK. <lb/></s>
  <s xml:id="echoid-s3332" xml:space="preserve">KC :</s>
  <s xml:id="echoid-s3333" xml:space="preserve">: NP x AN. </s>
  <s xml:id="echoid-s3334" xml:space="preserve">CNq. </s>
  <s xml:id="echoid-s3335" xml:space="preserve">componendóque AC. </s>
  <s xml:id="echoid-s3336" xml:space="preserve">KC :</s>
  <s xml:id="echoid-s3337" xml:space="preserve">: NP <lb/>
<anchor type="note" xlink:label="note-0077-01a" xlink:href="note-0077-01"/>
x AN + CNq. </s>
  <s xml:id="echoid-s3338" xml:space="preserve">CNq. </s>
  <s xml:id="echoid-s3339" xml:space="preserve">cum ſit igitur NP x AN = AP x AN <lb/>- ANq. </s>
  <s xml:id="echoid-s3340" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3341" xml:space="preserve">AP x AN = ACq - CNq; </s>
  <s xml:id="echoid-s3342" xml:space="preserve">adeóque NP <lb/>x AN + CNq = ACq - CNq - ANq + CNq; </s>
  <s xml:id="echoid-s3343" xml:space="preserve">= ACq <lb/>- ANq. </s>
  <s xml:id="echoid-s3344" xml:space="preserve">erit AC. </s>
  <s xml:id="echoid-s3345" xml:space="preserve">KC :</s>
  <s xml:id="echoid-s3346" xml:space="preserve">: ACq - ANq. </s>
  <s xml:id="echoid-s3347" xml:space="preserve">CNq : </s>
  <s xml:id="echoid-s3348" xml:space="preserve">Quod E. </s>
  <s xml:id="echoid-s3349" xml:space="preserve">D. <lb/></s>
  <s xml:id="echoid-s3350" xml:space="preserve">_Coroll_. </s>
  <s xml:id="echoid-s3351" xml:space="preserve">AK. </s>
  <s xml:id="echoid-s3352" xml:space="preserve">KC :</s>
  <s xml:id="echoid-s3353" xml:space="preserve">: AN x NP. </s>
  <s xml:id="echoid-s3354" xml:space="preserve">CNq.</s>
  <s xml:id="echoid-s3355" xml:space="preserve"/>
</p>
<div xml:id="echoid-div91" type="float" level="2" n="1">
<note position="right" xlink:label="note-0077-01" xlink:href="note-0077-01a" xml:space="preserve">Fig. 79, 80.</note>
</div>
<p>
  <s xml:id="echoid-s3356" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s3357" xml:space="preserve">Etiam hoc _Theorema_ ſubdemus: </s>
  <s xml:id="echoid-s3358" xml:space="preserve">Si fiat 2 CA. </s>
  <s xml:id="echoid-s3359" xml:space="preserve">CN :</s>
  <s xml:id="echoid-s3360" xml:space="preserve">: <lb/>CN. </s>
  <s xml:id="echoid-s3361" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3362" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3363" xml:space="preserve">2 CK. </s>
  <s xml:id="echoid-s3364" xml:space="preserve">CN :</s>
  <s xml:id="echoid-s3365" xml:space="preserve">: CN. </s>
  <s xml:id="echoid-s3366" xml:space="preserve">F; </s>
  <s xml:id="echoid-s3367" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3368" xml:space="preserve">ſumatur CQ = E + F; <lb/></s>
  <s xml:id="echoid-s3369" xml:space="preserve">erit ducta NQ ad CA perpendicularis. </s>
  <s xml:id="echoid-s3370" xml:space="preserve">vel reciprocè; </s>
  <s xml:id="echoid-s3371" xml:space="preserve">poſito quòd <lb/>ſit NQ ad CA perpendicularis; </s>
  <s xml:id="echoid-s3372" xml:space="preserve">erit CQ = E + F. </s>
  <s xml:id="echoid-s3373" xml:space="preserve">‖ Nam (ut <lb/>hoc poſterius oſtendamus) quoniam eſt 2 CA. </s>
  <s xml:id="echoid-s3374" xml:space="preserve">CN :</s>
  <s xml:id="echoid-s3375" xml:space="preserve">: CN. </s>
  <s xml:id="echoid-s3376" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3377" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s3378" xml:space="preserve">CN. </s>
  <s xml:id="echoid-s3379" xml:space="preserve">2 CK :</s>
  <s xml:id="echoid-s3380" xml:space="preserve">: F. </s>
  <s xml:id="echoid-s3381" xml:space="preserve">CN. </s>
  <s xml:id="echoid-s3382" xml:space="preserve">erit ex æquo perturbatè 2 CA. </s>
  <s xml:id="echoid-s3383" xml:space="preserve">2 CK <lb/>:</s>
  <s xml:id="echoid-s3384" xml:space="preserve">: F. </s>
  <s xml:id="echoid-s3385" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3386" xml:space="preserve">vel CA. </s>
  <s xml:id="echoid-s3387" xml:space="preserve">CK :</s>
  <s xml:id="echoid-s3388" xml:space="preserve">: F. </s>
  <s xml:id="echoid-s3389" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3390" xml:space="preserve">componendóque CA + CK. </s>
  <s xml:id="echoid-s3391" xml:space="preserve">CK <lb/>:</s>
  <s xml:id="echoid-s3392" xml:space="preserve">: F + E. </s>
  <s xml:id="echoid-s3393" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3394" xml:space="preserve">Porrò quoniam eſt ANq = ACq + CNq - 2 AC <lb/>x CQ; </s>
  <s xml:id="echoid-s3395" xml:space="preserve">erit 2 AC x CQ - CNq = ACq - ANq. </s>
  <s xml:id="echoid-s3396" xml:space="preserve">itaque <lb/>(juxta præcedentem) erit 2 AC x CQ - CNq. </s>
  <s xml:id="echoid-s3397" xml:space="preserve">CNQ :</s>
  <s xml:id="echoid-s3398" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s3399" xml:space="preserve"><lb/>CK. </s>
  <s xml:id="echoid-s3400" xml:space="preserve">hoc eſt ( ob CNq = 2 AC x E) 2 AC x CQ - 2 AC <lb/>x E. </s>
  <s xml:id="echoid-s3401" xml:space="preserve">2 AC x E :</s>
  <s xml:id="echoid-s3402" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s3403" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s3404" xml:space="preserve">hoc eſt CQ - E. </s>
  <s xml:id="echoid-s3405" xml:space="preserve">E :</s>
  <s xml:id="echoid-s3406" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s3407" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s3408" xml:space="preserve"><lb/>vel componendo CQ. </s>
  <s xml:id="echoid-s3409" xml:space="preserve">E :</s>
  <s xml:id="echoid-s3410" xml:space="preserve">: AC + CK. </s>
  <s xml:id="echoid-s3411" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s3412" xml:space="preserve">erat autem AC <lb/>+ CK. </s>
  <s xml:id="echoid-s3413" xml:space="preserve">CK :</s>
  <s xml:id="echoid-s3414" xml:space="preserve">: F + E. </s>
  <s xml:id="echoid-s3415" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3416" xml:space="preserve">ergò CQ = F + E: </s>
  <s xml:id="echoid-s3417" xml:space="preserve">Quod E. </s>
  <s xml:id="echoid-s3418" xml:space="preserve">D.</s>
  <s xml:id="echoid-s3419" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3420" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s3421" xml:space="preserve">Ex iſtis porrò deducetur, ſi dividatur Semidiameter BC in <lb/>Z, ut ſit AC. </s>
  <s xml:id="echoid-s3422" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s3423" xml:space="preserve">: CZ. </s>
  <s xml:id="echoid-s3424" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s3425" xml:space="preserve">punctum Z limes erit citra quem <lb/>(reſpectu centri C) nullus hujuſmodi reflexus axem decuſſabit. </s>
  <s xml:id="echoid-s3426" xml:space="preserve">Cu-<lb/>juſvis, inquam, radii AN eſto reflexus GN; </s>
  <s xml:id="echoid-s3427" xml:space="preserve">axi occurrens in K. <lb/></s>
  <s xml:id="echoid-s3428" xml:space="preserve">dico fore CK &amp;</s>
  <s xml:id="echoid-s3429" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3430" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s3431" xml:space="preserve">Nam ob hypotheſin (permutandóque) eſt AC. </s>
  <s xml:id="echoid-s3432" xml:space="preserve"><lb/>CZ :</s>
  <s xml:id="echoid-s3433" xml:space="preserve">: AB. </s>
  <s xml:id="echoid-s3434" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s3435" xml:space="preserve">igitur (antecedentes, &amp; </s>
  <s xml:id="echoid-s3436" xml:space="preserve">conſequentes copulan-<lb/>do) AC. </s>
  <s xml:id="echoid-s3437" xml:space="preserve">CZ :</s>
  <s xml:id="echoid-s3438" xml:space="preserve">: AC + AB. </s>
  <s xml:id="echoid-s3439" xml:space="preserve">CB. </s>
  <s xml:id="echoid-s3440" xml:space="preserve">quare ( poſterioris hujuſce <lb/>rationis utrumque terminum in æquales AC - AB, &amp; </s>
  <s xml:id="echoid-s3441" xml:space="preserve">BC ducen-<lb/>do) erit AC. </s>
  <s xml:id="echoid-s3442" xml:space="preserve">CZ :</s>
  <s xml:id="echoid-s3443" xml:space="preserve">: ACq - ABq. </s>
  <s xml:id="echoid-s3444" xml:space="preserve">CBq. </s>
  <s xml:id="echoid-s3445" xml:space="preserve">eſt autem ACq <lb/>- ABq &amp;</s>
  <s xml:id="echoid-s3446" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3447" xml:space="preserve">ACq - ANq; </s>
  <s xml:id="echoid-s3448" xml:space="preserve">adeóque ACq - ABq CBq. </s>
  <s xml:id="echoid-s3449" xml:space="preserve"><lb/>&amp;</s>
  <s xml:id="echoid-s3450" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3451" xml:space="preserve">ACq - ANq. </s>
  <s xml:id="echoid-s3452" xml:space="preserve">CBq :</s>
  <s xml:id="echoid-s3453" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s3454" xml:space="preserve">CK (è mox oſtenſis hoc) qua-<lb/>propter erit AC. </s>
  <s xml:id="echoid-s3455" xml:space="preserve">CZ &amp;</s>
  <s xml:id="echoid-s3456" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3457" xml:space="preserve">AC. </s>
  <s xml:id="echoid-s3458" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s3459" xml:space="preserve">indéque CK &amp;</s>
  <s xml:id="echoid-s3460" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3461" xml:space="preserve">CZ: </s>
  <s xml:id="echoid-s3462" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0077-02a" xlink:href="note-0077-02"/>
Q. </s>
  <s xml:id="echoid-s3463" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3464" xml:space="preserve">D.</s>
  <s xml:id="echoid-s3465" xml:space="preserve"/>
</p>
<div xml:id="echoid-div92" type="float" level="2" n="2">
<note position="right" xlink:label="note-0077-02" xlink:href="note-0077-02a" xml:space="preserve">Fig. 81, 82.</note>
</div>
<p>
  <s xml:id="echoid-s3466" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s3467" xml:space="preserve">Aliter hoc idem; </s>
  <s xml:id="echoid-s3468" xml:space="preserve">ut quibuſdam fortaſſe videbitur, minùs <lb/>involutè: </s>
  <s xml:id="echoid-s3469" xml:space="preserve">per N ducatur VT circulum contingens. </s>
  <s xml:id="echoid-s3470" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3471" xml:space="preserve">quoniam NT
<pb o="60" file="0078" n="78" rhead=""/>
biſecat angulum ANK; </s>
  <s xml:id="echoid-s3472" xml:space="preserve">erit AN. </s>
  <s xml:id="echoid-s3473" xml:space="preserve">NK :</s>
  <s xml:id="echoid-s3474" xml:space="preserve">: AT. </s>
  <s xml:id="echoid-s3475" xml:space="preserve">TK. </s>
  <s xml:id="echoid-s3476" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s3477" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3478" xml:space="preserve">AB. <lb/></s>
  <s xml:id="echoid-s3479" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s3480" xml:space="preserve">quare BZ. </s>
  <s xml:id="echoid-s3481" xml:space="preserve">AB + AN. </s>
  <s xml:id="echoid-s3482" xml:space="preserve">NK &amp;</s>
  <s xml:id="echoid-s3483" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3484" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s3485" xml:space="preserve">AB + AB. </s>
  <s xml:id="echoid-s3486" xml:space="preserve">BK <lb/>(communem adſciſcendo rationem BZ ad AB). </s>
  <s xml:id="echoid-s3487" xml:space="preserve">eſt autem BZ. </s>
  <s xml:id="echoid-s3488" xml:space="preserve"><lb/>AB + AN. </s>
  <s xml:id="echoid-s3489" xml:space="preserve">NK = CZ. </s>
  <s xml:id="echoid-s3490" xml:space="preserve">AC + AC. </s>
  <s xml:id="echoid-s3491" xml:space="preserve">CK = CZ. </s>
  <s xml:id="echoid-s3492" xml:space="preserve">CK &amp; </s>
  <s xml:id="echoid-s3493" xml:space="preserve"><lb/>BZ. </s>
  <s xml:id="echoid-s3494" xml:space="preserve">AB + AB. </s>
  <s xml:id="echoid-s3495" xml:space="preserve">BK = BZ. </s>
  <s xml:id="echoid-s3496" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s3497" xml:space="preserve">ergo CZ. </s>
  <s xml:id="echoid-s3498" xml:space="preserve">CK &amp;</s>
  <s xml:id="echoid-s3499" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3500" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s3501" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s3502" xml:space="preserve"><lb/>permutandóque CZ. </s>
  <s xml:id="echoid-s3503" xml:space="preserve">BZ &amp;</s>
  <s xml:id="echoid-s3504" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3505" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s3506" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s3507" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s3508" xml:space="preserve">componendo CB. </s>
  <s xml:id="echoid-s3509" xml:space="preserve"><lb/>BZ &amp;</s>
  <s xml:id="echoid-s3510" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3511" xml:space="preserve">CB. </s>
  <s xml:id="echoid-s3512" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s3513" xml:space="preserve">ideóque BZ &amp;</s>
  <s xml:id="echoid-s3514" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3515" xml:space="preserve">BK; </s>
  <s xml:id="echoid-s3516" xml:space="preserve">quare punctum Z centro <lb/>propinquius eſt, quàm ipſum K: </s>
  <s xml:id="echoid-s3517" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s3518" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3519" xml:space="preserve">D.</s>
  <s xml:id="echoid-s3520" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3521" xml:space="preserve">_Coroll_. </s>
  <s xml:id="echoid-s3522" xml:space="preserve">Hinc ſi puncta Z, ζ fuerint limites punctorum radiantium <lb/>A, _a_ (quorum A fit à fpeculo remotiùs, quàm _a_) erit CZ &amp;</s>
  <s xml:id="echoid-s3523" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3524" xml:space="preserve">C ζ. <lb/></s>
  <s xml:id="echoid-s3525" xml:space="preserve">Nam eſt BC. </s>
  <s xml:id="echoid-s3526" xml:space="preserve">AB &amp;</s>
  <s xml:id="echoid-s3527" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3528" xml:space="preserve">BC, _a_ B. </s>
  <s xml:id="echoid-s3529" xml:space="preserve">adeóque compoſitè AC. </s>
  <s xml:id="echoid-s3530" xml:space="preserve">AB &amp;</s>
  <s xml:id="echoid-s3531" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3532" xml:space="preserve"><lb/>_a_ C. </s>
  <s xml:id="echoid-s3533" xml:space="preserve">_a_ B. </s>
  <s xml:id="echoid-s3534" xml:space="preserve">hoc eſt CZ. </s>
  <s xml:id="echoid-s3535" xml:space="preserve">BZ &amp;</s>
  <s xml:id="echoid-s3536" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3537" xml:space="preserve">C ζ. </s>
  <s xml:id="echoid-s3538" xml:space="preserve">B ζ. </s>
  <s xml:id="echoid-s3539" xml:space="preserve">quare componendo <lb/>BC. </s>
  <s xml:id="echoid-s3540" xml:space="preserve">BZ &amp;</s>
  <s xml:id="echoid-s3541" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3542" xml:space="preserve">BC. </s>
  <s xml:id="echoid-s3543" xml:space="preserve">B ζ. </s>
  <s xml:id="echoid-s3544" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3545" xml:space="preserve">indè BZ &amp;</s>
  <s xml:id="echoid-s3546" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3547" xml:space="preserve">B ζ.</s>
  <s xml:id="echoid-s3548" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3549" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s3550" xml:space="preserve">Porrò, conſectatur è præmiſſis, quòd ſi duorum quorumvis <lb/>incidentium AN, AR reflexi GN, HR axem interſecent punctis <lb/>K, L; </s>
  <s xml:id="echoid-s3551" xml:space="preserve">erit CL, CK :</s>
  <s xml:id="echoid-s3552" xml:space="preserve">: ACq - ANq. </s>
  <s xml:id="echoid-s3553" xml:space="preserve">ACq - ARq. </s>
  <s xml:id="echoid-s3554" xml:space="preserve">‖ Nam <lb/>quoniam eſt AC. </s>
  <s xml:id="echoid-s3555" xml:space="preserve">CK :</s>
  <s xml:id="echoid-s3556" xml:space="preserve">: ACq - ANq. </s>
  <s xml:id="echoid-s3557" xml:space="preserve">CBq. </s>
  <s xml:id="echoid-s3558" xml:space="preserve">itémque CL. <lb/></s>
  <s xml:id="echoid-s3559" xml:space="preserve">AC :</s>
  <s xml:id="echoid-s3560" xml:space="preserve">: CBq. </s>
  <s xml:id="echoid-s3561" xml:space="preserve">ACq - ARq. </s>
  <s xml:id="echoid-s3562" xml:space="preserve">erit ex æquo perturbatè CL. </s>
  <s xml:id="echoid-s3563" xml:space="preserve"><lb/>CK :</s>
  <s xml:id="echoid-s3564" xml:space="preserve">: ACq - ANq. </s>
  <s xml:id="echoid-s3565" xml:space="preserve">ACq - ARq.</s>
  <s xml:id="echoid-s3566" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3567" xml:space="preserve">X. </s>
  <s xml:id="echoid-s3568" xml:space="preserve">Simili planè diſcurſu, ſi fuerit AC. </s>
  <s xml:id="echoid-s3569" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s3570" xml:space="preserve">: CZ. </s>
  <s xml:id="echoid-s3571" xml:space="preserve">ZB. <lb/></s>
  <s xml:id="echoid-s3572" xml:space="preserve">erit CZ. </s>
  <s xml:id="echoid-s3573" xml:space="preserve">CK :</s>
  <s xml:id="echoid-s3574" xml:space="preserve">: ACq - ANq. </s>
  <s xml:id="echoid-s3575" xml:space="preserve">ACq - ABq. </s>
  <s xml:id="echoid-s3576" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3577" xml:space="preserve">CL. </s>
  <s xml:id="echoid-s3578" xml:space="preserve">CZ :</s>
  <s xml:id="echoid-s3579" xml:space="preserve">: <lb/>ACq - ARq. </s>
  <s xml:id="echoid-s3580" xml:space="preserve">ACq - ABq.</s>
  <s xml:id="echoid-s3581" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3582" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s3583" xml:space="preserve">Hinc perſpicuum eſt obliquioris reflexi concurſum à centro <lb/>magìs elongari quam rectioris; </s>
  <s xml:id="echoid-s3584" xml:space="preserve">quod nempe ſit CL &amp;</s>
  <s xml:id="echoid-s3585" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3586" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s3587" xml:space="preserve">Cùm <lb/>enim ſit ACq - ANq &amp;</s>
  <s xml:id="echoid-s3588" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3589" xml:space="preserve">ACq - ARq; </s>
  <s xml:id="echoid-s3590" xml:space="preserve">erit CL &amp;</s>
  <s xml:id="echoid-s3591" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3592" xml:space="preserve">CK.</s>
  <s xml:id="echoid-s3593" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3594" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s3595" xml:space="preserve">Hinc neceſſariò duo quilibet ad eaſdem axis partes incidentium <lb/>reflexi (quales NK, RL) ſeſe priùs quàm axem interſecabunt, puta <lb/>ad X. </s>
  <s xml:id="echoid-s3596" xml:space="preserve">quo poſito.</s>
  <s xml:id="echoid-s3597" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3598" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s3599" xml:space="preserve">Adnotari poteſt angulum GXH vel KXL (à reflexis oc-<lb/>currentibus incluſum) æquari angulo NCR unà cum differentia angu-<lb/>lorum incidentiæ; </s>
  <s xml:id="echoid-s3600" xml:space="preserve">vel, duplo angulo NCR unà cum ang. </s>
  <s xml:id="echoid-s3601" xml:space="preserve">NAR. </s>
  <s xml:id="echoid-s3602" xml:space="preserve">‖ Etenim <lb/>ang. </s>
  <s xml:id="echoid-s3603" xml:space="preserve">KXL = ang. </s>
  <s xml:id="echoid-s3604" xml:space="preserve">ALR - AKN = ang. </s>
  <s xml:id="echoid-s3605" xml:space="preserve">ACR + CRL -: </s>
  <s xml:id="echoid-s3606" xml:space="preserve">ang. <lb/></s>
  <s xml:id="echoid-s3607" xml:space="preserve">
<anchor type="note" xlink:label="note-0078-01a" xlink:href="note-0078-01"/>
ACN + CNK = ang. </s>
  <s xml:id="echoid-s3608" xml:space="preserve">ACR - ACN + : </s>
  <s xml:id="echoid-s3609" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s3610" xml:space="preserve">CRL -<lb/>CNK = ang NCR +: </s>
  <s xml:id="echoid-s3611" xml:space="preserve">ang CRS - CNP. </s>
  <s xml:id="echoid-s3612" xml:space="preserve">‖ Quinetiam ang. <lb/></s>
  <s xml:id="echoid-s3613" xml:space="preserve">CRS - CNP = ang. </s>
  <s xml:id="echoid-s3614" xml:space="preserve">RCA + CAR -: </s>
  <s xml:id="echoid-s3615" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s3616" xml:space="preserve">NCA +
<pb o="61" file="0079" n="79" rhead=""/>
CAN = ang. </s>
  <s xml:id="echoid-s3617" xml:space="preserve">NCR + NAR. </s>
  <s xml:id="echoid-s3618" xml:space="preserve">itaque rurſus ang. </s>
  <s xml:id="echoid-s3619" xml:space="preserve">KXL = <lb/>2 ang. </s>
  <s xml:id="echoid-s3620" xml:space="preserve">NCR + ang. </s>
  <s xml:id="echoid-s3621" xml:space="preserve">NAR. </s>
  <s xml:id="echoid-s3622" xml:space="preserve">liquent igitur quæ propoſita ſunt; <lb/></s>
  <s xml:id="echoid-s3623" xml:space="preserve">in uſum (ſi fortè) ſequentium. </s>
  <s xml:id="echoid-s3624" xml:space="preserve">pro quibus itidem hæc proponenda <lb/>ſunt.</s>
  <s xml:id="echoid-s3625" xml:space="preserve"/>
</p>
<div xml:id="echoid-div93" type="float" level="2" n="3">
<note position="left" xlink:label="note-0078-01" xlink:href="note-0078-01a" xml:space="preserve">Fig. 83.</note>
</div>
<p>
  <s xml:id="echoid-s3626" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s3627" xml:space="preserve">Etiam palàm eſt è dictis ipſos reflexos GN, HR directè <lb/>
<anchor type="note" xlink:label="note-0079-01a" xlink:href="note-0079-01"/>
procurrentes à ſe divergere; </s>
  <s xml:id="echoid-s3628" xml:space="preserve">adeóque duntaxat unum hujuſmodi re-<lb/>flexum oculi centrum tranſire; </s>
  <s xml:id="echoid-s3629" xml:space="preserve">conſequentèr &amp; </s>
  <s xml:id="echoid-s3630" xml:space="preserve">puncti A tantùm u-<lb/>nam à convexo ſpeculo imaginem exhiberi.</s>
  <s xml:id="echoid-s3631" xml:space="preserve"/>
</p>
<div xml:id="echoid-div94" type="float" level="2" n="4">
<note position="right" xlink:label="note-0079-01" xlink:href="note-0079-01a" xml:space="preserve">Fig. 83.</note>
</div>
<p>
  <s xml:id="echoid-s3632" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s3633" xml:space="preserve">_Lemmatia_ 1. </s>
  <s xml:id="echoid-s3634" xml:space="preserve">Sint quæcunque tria quanta A, P, C; </s>
  <s xml:id="echoid-s3635" xml:space="preserve">primó-<lb/>que ſit A. </s>
  <s xml:id="echoid-s3636" xml:space="preserve">B &amp;</s>
  <s xml:id="echoid-s3637" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3638" xml:space="preserve">B. </s>
  <s xml:id="echoid-s3639" xml:space="preserve">C; </s>
  <s xml:id="echoid-s3640" xml:space="preserve">dico fore A + C &amp;</s>
  <s xml:id="echoid-s3641" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3642" xml:space="preserve">2 B. </s>
  <s xml:id="echoid-s3643" xml:space="preserve">ponatur enim <lb/>fore A. </s>
  <s xml:id="echoid-s3644" xml:space="preserve">B :</s>
  <s xml:id="echoid-s3645" xml:space="preserve">: B. </s>
  <s xml:id="echoid-s3646" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3647" xml:space="preserve">erit ergò A + E &amp;</s>
  <s xml:id="echoid-s3648" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3649" xml:space="preserve">2 B. </s>
  <s xml:id="echoid-s3650" xml:space="preserve">quinetiam erit ergò <lb/>B. </s>
  <s xml:id="echoid-s3651" xml:space="preserve">E &amp;</s>
  <s xml:id="echoid-s3652" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3653" xml:space="preserve">B. </s>
  <s xml:id="echoid-s3654" xml:space="preserve">C adeóque C &amp;</s>
  <s xml:id="echoid-s3655" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3656" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3657" xml:space="preserve">ergo magis A + C &amp;</s>
  <s xml:id="echoid-s3658" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3659" xml:space="preserve">2 B.</s>
  <s xml:id="echoid-s3660" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3661" xml:space="preserve">2. </s>
  <s xml:id="echoid-s3662" xml:space="preserve">Sit (iiſdem adhibitis quantis) ſecundò A + C &amp;</s>
  <s xml:id="echoid-s3663" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3664" xml:space="preserve">2 B. </s>
  <s xml:id="echoid-s3665" xml:space="preserve">dico <lb/>fore A. </s>
  <s xml:id="echoid-s3666" xml:space="preserve">B &amp;</s>
  <s xml:id="echoid-s3667" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3668" xml:space="preserve">B. </s>
  <s xml:id="echoid-s3669" xml:space="preserve">C. </s>
  <s xml:id="echoid-s3670" xml:space="preserve">nam ſive dicatur eſſe A. </s>
  <s xml:id="echoid-s3671" xml:space="preserve">B :</s>
  <s xml:id="echoid-s3672" xml:space="preserve">: B. </s>
  <s xml:id="echoid-s3673" xml:space="preserve">C. </s>
  <s xml:id="echoid-s3674" xml:space="preserve">vel A. </s>
  <s xml:id="echoid-s3675" xml:space="preserve">B <lb/>&amp;</s>
  <s xml:id="echoid-s3676" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3677" xml:space="preserve">B. </s>
  <s xml:id="echoid-s3678" xml:space="preserve">C. </s>
  <s xml:id="echoid-s3679" xml:space="preserve">ſequetur utrobique fore A + C &amp;</s>
  <s xml:id="echoid-s3680" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3681" xml:space="preserve">2 B; </s>
  <s xml:id="echoid-s3682" xml:space="preserve">contra hypothe-<lb/>ſin. </s>
  <s xml:id="echoid-s3683" xml:space="preserve">itaque potiùs eſt A. </s>
  <s xml:id="echoid-s3684" xml:space="preserve">B &amp;</s>
  <s xml:id="echoid-s3685" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3686" xml:space="preserve">B. </s>
  <s xml:id="echoid-s3687" xml:space="preserve">C.</s>
  <s xml:id="echoid-s3688" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3689" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s3690" xml:space="preserve">Etiam hoc adjungo. </s>
  <s xml:id="echoid-s3691" xml:space="preserve">Si duo ſumantur ad eaſdem axi partes <lb/>
<anchor type="note" xlink:label="note-0079-02a" xlink:href="note-0079-02"/>
(circulíque convexâ parte comprehenſi) ſibimet æquales arcus NR, <lb/>R X; </s>
  <s xml:id="echoid-s3692" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3693" xml:space="preserve">ducantur rectæ AN, AR, AX; </s>
  <s xml:id="echoid-s3694" xml:space="preserve">erit ANq + AXq <lb/>&amp;</s>
  <s xml:id="echoid-s3695" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3696" xml:space="preserve">2 ARq.</s>
  <s xml:id="echoid-s3697" xml:space="preserve"/>
</p>
<div xml:id="echoid-div95" type="float" level="2" n="5">
<note position="right" xlink:label="note-0079-02" xlink:href="note-0079-02a" xml:space="preserve">Fig. 84.</note>
</div>
<p>
  <s xml:id="echoid-s3698" xml:space="preserve">Nam ducantur CN, CR, CX; </s>
  <s xml:id="echoid-s3699" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3700" xml:space="preserve">demittantur ad AC perpen-<lb/>diculares NE, RF, X G; </s>
  <s xml:id="echoid-s3701" xml:space="preserve">ſint item NP, RQ ad AC parallelæ <lb/>ducantúrque ſubtenſæ NR, RX. </s>
  <s xml:id="echoid-s3702" xml:space="preserve">; </s>
  <s xml:id="echoid-s3703" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3704" xml:space="preserve">quoniam ang. </s>
  <s xml:id="echoid-s3705" xml:space="preserve">RXQ &amp;</s>
  <s xml:id="echoid-s3706" xml:space="preserve">gt; <lb/></s>
  <s xml:id="echoid-s3707" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s3708" xml:space="preserve">NRP; </s>
  <s xml:id="echoid-s3709" xml:space="preserve">patet eſſe R X. </s>
  <s xml:id="echoid-s3710" xml:space="preserve">RQ &amp;</s>
  <s xml:id="echoid-s3711" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3712" xml:space="preserve">NR. </s>
  <s xml:id="echoid-s3713" xml:space="preserve">NP; </s>
  <s xml:id="echoid-s3714" xml:space="preserve">adeóque cum <lb/>RX = NR, erit RQ &amp;</s>
  <s xml:id="echoid-s3715" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3716" xml:space="preserve">NP; </s>
  <s xml:id="echoid-s3717" xml:space="preserve">hoc eſt FG &amp;</s>
  <s xml:id="echoid-s3718" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3719" xml:space="preserve">EF. </s>
  <s xml:id="echoid-s3720" xml:space="preserve">ergò 2 CF <lb/>&amp;</s>
  <s xml:id="echoid-s3721" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3722" xml:space="preserve">CE + CG; </s>
  <s xml:id="echoid-s3723" xml:space="preserve">unde 4 AC x CF &amp;</s>
  <s xml:id="echoid-s3724" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3725" xml:space="preserve">2 AC x CE + 2 AC x <lb/>CG atqui eſt ANq = ACq + CNq - 2 AC x CE. </s>
  <s xml:id="echoid-s3726" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3727" xml:space="preserve">AXq <lb/> = ACq + CNq - 2 AC x CG. </s>
  <s xml:id="echoid-s3728" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3729" xml:space="preserve">2 ARq = 2 ACq + <lb/>2 CNq - 4 AC x CF. </s>
  <s xml:id="echoid-s3730" xml:space="preserve">ergo ANq + AXq &amp;</s>
  <s xml:id="echoid-s3731" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3732" xml:space="preserve">2 ARq.</s>
  <s xml:id="echoid-s3733" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3734" xml:space="preserve">Addo, ſequentium gratià, ſi punctum A ſumatur ad alteras (inſra <lb/>centrum) partes; </s>
  <s xml:id="echoid-s3735" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3736" xml:space="preserve">reliqua ſimiliter apparentur; </s>
  <s xml:id="echoid-s3737" xml:space="preserve">fore contrà, tum <lb/>ANq + AXq &amp;</s>
  <s xml:id="echoid-s3738" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3739" xml:space="preserve">2 ARq. </s>
  <s xml:id="echoid-s3740" xml:space="preserve">nam in eo caſu eſt ANq + AXq <lb/> = 2 ACq + 2 CNq + 2 AC x CE + 2 AC x CG. </s>
  <s xml:id="echoid-s3741" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3742" xml:space="preserve"><lb/>2 ARq = 2 ACq + 2 CNq + 4 AC x CF. </s>
  <s xml:id="echoid-s3743" xml:space="preserve">unde liquet pro-<lb/>poſitum.</s>
  <s xml:id="echoid-s3744" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3745" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s3746" xml:space="preserve">Sint jam ad eaſdem axis partes duo quilibet æquales arcus
<pb o="62" file="0080" n="80" rhead=""/>
NR, RX; </s>
  <s xml:id="echoid-s3747" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3748" xml:space="preserve">incidentinm AN, AR, AX reflexi GN, HR, IX <lb/>
<anchor type="note" xlink:label="note-0080-01a" xlink:href="note-0080-01"/>
axi occurrant producti punctis K, L, M; </s>
  <s xml:id="echoid-s3749" xml:space="preserve">erit intervallum ML ab <lb/>obliquiorum occurſibus concluſum majus ipſo LK rectiorum occurſi-<lb/>bus intercepto.</s>
  <s xml:id="echoid-s3750" xml:space="preserve"/>
</p>
<div xml:id="echoid-div96" type="float" level="2" n="6">
<note position="left" xlink:label="note-0080-01" xlink:href="note-0080-01a" xml:space="preserve">Fig. 85.</note>
</div>
<p>
  <s xml:id="echoid-s3751" xml:space="preserve">Nam quoniam eſt ANq + AXq &amp;</s>
  <s xml:id="echoid-s3752" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3753" xml:space="preserve">2 ARq. </s>
  <s xml:id="echoid-s3754" xml:space="preserve">erit 2 ACq <lb/>- ANq - AXq &amp;</s>
  <s xml:id="echoid-s3755" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3756" xml:space="preserve">2 ACq - 2 ARq. </s>
  <s xml:id="echoid-s3757" xml:space="preserve">adeóque ACq -<lb/>- AXq. </s>
  <s xml:id="echoid-s3758" xml:space="preserve">ACq - ARq &amp;</s>
  <s xml:id="echoid-s3759" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3760" xml:space="preserve">ACq - ARq. </s>
  <s xml:id="echoid-s3761" xml:space="preserve">ACq - ANq. <lb/></s>
  <s xml:id="echoid-s3762" xml:space="preserve">hoc eſt, è præmonſtratis, CL. </s>
  <s xml:id="echoid-s3763" xml:space="preserve">CM &amp;</s>
  <s xml:id="echoid-s3764" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3765" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s3766" xml:space="preserve">CL; </s>
  <s xml:id="echoid-s3767" xml:space="preserve">vel inversè <lb/>CM. </s>
  <s xml:id="echoid-s3768" xml:space="preserve">CL &amp;</s>
  <s xml:id="echoid-s3769" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3770" xml:space="preserve">CL. </s>
  <s xml:id="echoid-s3771" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s3772" xml:space="preserve">quapropter erit CM + CK. </s>
  <s xml:id="echoid-s3773" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s3774" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3775" xml:space="preserve">2 CL. </s>
  <s xml:id="echoid-s3776" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s3777" xml:space="preserve">ideò CM - CL &amp;</s>
  <s xml:id="echoid-s3778" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3779" xml:space="preserve">CL - CK hoc eſt ML &amp;</s>
  <s xml:id="echoid-s3780" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3781" xml:space="preserve">LK : </s>
  <s xml:id="echoid-s3782" xml:space="preserve"><lb/>Q E. </s>
  <s xml:id="echoid-s3783" xml:space="preserve">D.</s>
  <s xml:id="echoid-s3784" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3785" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s3786" xml:space="preserve">Hinc conſtat, etiam in hac hypotheſi, rectiùs incidentem <lb/>lucem â reflectione magìs inſpiſſari; </s>
  <s xml:id="echoid-s3787" xml:space="preserve">ſeu ſpatio verſus limitem Z arcti-<lb/>ore conſtringi.</s>
  <s xml:id="echoid-s3788" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3789" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s3790" xml:space="preserve">Quin ab his demum omnibus colligitur, ſi uſpiam in axe <lb/>(velut ad O) conſtituatur oculi centrum, quod punctum A neceſſa-<lb/>
<anchor type="note" xlink:label="note-0080-02a" xlink:href="note-0080-02"/>
riò circa limitem Z apparebit. </s>
  <s xml:id="echoid-s3791" xml:space="preserve">Etenim (prorſus ut in præcedente <lb/>quoad radios ab infinitè diſſito puncto manantes hypotheſi) ab axis <lb/>illi puncto adjacente parte radii cùm copioſiores, tum axi viciniores, <lb/>oculóque rectiores, efficaciâ proinde præpollentes, nec non qui <lb/>faciliùs re-adunentur, provenire videntur. </s>
  <s xml:id="echoid-s3792" xml:space="preserve">quæ nempe cuncta ſimul <lb/>ac emergentem propoſiti conſequentiam abunde, puto, dedimus <lb/>enucleata. </s>
  <s xml:id="echoid-s3793" xml:space="preserve">Succedit ut hâc parte defuncti pro viſu extra radiationis <lb/>axem collocato itidem imaginis ſedem definiamus. </s>
  <s xml:id="echoid-s3794" xml:space="preserve">veruntamen hæc, <lb/>quanquam haud ità quantitate multa , pro rei tamen obſcuritate for-<lb/>taſſis nimia videbuntur. </s>
  <s xml:id="echoid-s3795" xml:space="preserve">itaque jam opportunum autumo deſiſtere.</s>
  <s xml:id="echoid-s3796" xml:space="preserve"/>
</p>
<div xml:id="echoid-div97" type="float" level="2" n="7">
<note position="left" xlink:label="note-0080-02" xlink:href="note-0080-02a" xml:space="preserve">Fig. 85.</note>
</div>
<pb o="63" file="0081" n="81"/>
</div>
<div xml:id="echoid-div99" type="section" level="1" n="18">
<head xml:id="echoid-head21" xml:space="preserve"><emph style="sc">Lect.</emph> IX.</head>
<p>
  <s xml:id="echoid-s3797" xml:space="preserve">I. </s>
  <s xml:id="echoid-s3798" xml:space="preserve">Q Ualiter in obverſum Speculi circularis convexum finitè di-<lb/>ſtans punctum radiat, &amp; </s>
  <s xml:id="echoid-s3799" xml:space="preserve">ubi loci adparet oculo in recta con-<lb/>ſtituto per ipſum radians &amp; </s>
  <s xml:id="echoid-s3800" xml:space="preserve">ſpeculi centrum trajecta poſtremo con-<lb/>niſi demonſtrare; </s>
  <s xml:id="echoid-s3801" xml:space="preserve">nunc idem quoad aſpectum aliàs ubicunque ſitum <lb/>aggredimur expiſcari. </s>
  <s xml:id="echoid-s3802" xml:space="preserve">quò primum attinet ut rectam inveſtigemus, <lb/>in qua conſiſtet Imago; </s>
  <s xml:id="echoid-s3803" xml:space="preserve">tum ut punctum ejus in iſta recta præciſum <lb/>determinemus. </s>
  <s xml:id="echoid-s3804" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3805" xml:space="preserve">primo quidem negotio ſatisfactum erit hujuſmodi <lb/>_Prob@ema_ conficiendo; </s>
  <s xml:id="echoid-s3806" xml:space="preserve">quod (ſequentium quoque gratiâ) genera-<lb/>tim proponimus.</s>
  <s xml:id="echoid-s3807" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3808" xml:space="preserve">II. </s>
  <s xml:id="echoid-s3809" xml:space="preserve">_Dato circulo reflectente_ (cujus centrum C) _datiſque binis pun-_ <lb/>_ctis; </s>
  <s xml:id="echoid-s3810" xml:space="preserve">ab horum uno recta ducatur, cujus rtflexus per alterum tran-_ <lb/>_ſeat._</s>
  <s xml:id="echoid-s3811" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3812" xml:space="preserve">1. </s>
  <s xml:id="echoid-s3813" xml:space="preserve">Si data puncta (puta A, X) ſint ambo in circuli peripheria, <lb/>
<anchor type="note" xlink:label="note-0081-01a" xlink:href="note-0081-01"/>
manifeſtum eſt biſecto arcu AX in N, connexiſque ſubtenſis NA, <lb/>N X, rectas NA, NX ſibi mutuò reflexas fore; </s>
  <s xml:id="echoid-s3814" xml:space="preserve">ſeu, junctâ CN, <lb/>angulum CNXangulo CNA æquari.</s>
  <s xml:id="echoid-s3815" xml:space="preserve"/>
</p>
<div xml:id="echoid-div99" type="float" level="2" n="1">
<note position="right" xlink:label="note-0081-01" xlink:href="note-0081-01a" xml:space="preserve">Fig. 86.</note>
</div>
<p>
  <s xml:id="echoid-s3816" xml:space="preserve">2. </s>
  <s xml:id="echoid-s3817" xml:space="preserve">Etiam ſi datorum unum (X) in circumferentia ponatur; </s>
  <s xml:id="echoid-s3818" xml:space="preserve">liquet, <lb/>
<anchor type="note" xlink:label="note-0081-02a" xlink:href="note-0081-02"/>
connexis AX, CX, factóque angulo CXH = CXA, ſore XA, <lb/>XH alterum alterius reflexum.</s>
  <s xml:id="echoid-s3819" xml:space="preserve"/>
</p>
<div xml:id="echoid-div100" type="float" level="2" n="2">
<note position="right" xlink:label="note-0081-02" xlink:href="note-0081-02a" xml:space="preserve">Fig. 87.</note>
</div>
<p>
  <s xml:id="echoid-s3820" xml:space="preserve">3. </s>
  <s xml:id="echoid-s3821" xml:space="preserve">Item ſi data puncta (A, X) æqualiter à centro diſtent; </s>
  <s xml:id="echoid-s3822" xml:space="preserve">con-<lb/>
<anchor type="note" xlink:label="note-0081-03a" xlink:href="note-0081-03"/>
nexis rectis AC, XC, biſectóque angulo XCA à recta CN circu-<lb/>lum reflectentem interſecante ad N; </s>
  <s xml:id="echoid-s3823" xml:space="preserve">perſpicuum eſt conjunctas rectas <lb/>AN, XN, invicem in ſe reflecti; </s>
  <s xml:id="echoid-s3824" xml:space="preserve">vel angulum CNXipſi CNA <lb/>æquari.</s>
  <s xml:id="echoid-s3825" xml:space="preserve"/>
</p>
<div xml:id="echoid-div101" type="float" level="2" n="3">
<note position="right" xlink:label="note-0081-03" xlink:href="note-0081-03a" xml:space="preserve">Fig. 83.</note>
</div>
<p>
  <s xml:id="echoid-s3826" xml:space="preserve">III. </s>
  <s xml:id="echoid-s3827" xml:space="preserve">4. </s>
  <s xml:id="echoid-s3828" xml:space="preserve">Si puncta data (puta jam A, K) ambo exiſtant in recta per <lb/>reflectentis centrum tranſeunte (nempe AB KC.)</s>
  <s xml:id="echoid-s3829" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3830" xml:space="preserve">I. </s>
  <s xml:id="echoid-s3831" xml:space="preserve">Fiat CK. </s>
  <s xml:id="echoid-s3832" xml:space="preserve">AC:</s>
  <s xml:id="echoid-s3833" xml:space="preserve">: CB. </s>
  <s xml:id="echoid-s3834" xml:space="preserve">T. </s>
  <s xml:id="echoid-s3835" xml:space="preserve">ac inter CB, &amp; </s>
  <s xml:id="echoid-s3836" xml:space="preserve">T ſit proportione <lb/>
<anchor type="note" xlink:label="note-0081-04a" xlink:href="note-0081-04"/>
media V (unde CBq. </s>
  <s xml:id="echoid-s3837" xml:space="preserve">Vq:</s>
  <s xml:id="echoid-s3838" xml:space="preserve">: CB. </s>
  <s xml:id="echoid-s3839" xml:space="preserve">T:</s>
  <s xml:id="echoid-s3840" xml:space="preserve">: CK. </s>
  <s xml:id="echoid-s3841" xml:space="preserve">AC). </s>
  <s xml:id="echoid-s3842" xml:space="preserve">tum centro A, <lb/>intervallo √:</s>
  <s xml:id="echoid-s3843" xml:space="preserve">: ACq - Vq. </s>
  <s xml:id="echoid-s3844" xml:space="preserve">deſcribatur circulus reflectentem
<pb o="64" file="0082" n="82" rhead=""/>
fecans in N; </s>
  <s xml:id="echoid-s3845" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3846" xml:space="preserve">per N ducatur KN G; </s>
  <s xml:id="echoid-s3847" xml:space="preserve">hæc ipſius AN reflexa <lb/>erit.</s>
  <s xml:id="echoid-s3848" xml:space="preserve"/>
</p>
<div xml:id="echoid-div102" type="float" level="2" n="4">
<note position="right" xlink:label="note-0081-04" xlink:href="note-0081-04a" xml:space="preserve">Fig. 8@.</note>
</div>
<p>
  <s xml:id="echoid-s3849" xml:space="preserve">Nam ob ANq = ACq - Vq. </s>
  <s xml:id="echoid-s3850" xml:space="preserve">erit Vq = ACq - ANq. <lb/></s>
  <s xml:id="echoid-s3851" xml:space="preserve">
<anchor type="note" xlink:label="note-0082-01a" xlink:href="note-0082-01"/>
adeóque CBq. </s>
  <s xml:id="echoid-s3852" xml:space="preserve">ACq - ANq:</s>
  <s xml:id="echoid-s3853" xml:space="preserve">: (CBq. </s>
  <s xml:id="echoid-s3854" xml:space="preserve">Vq:</s>
  <s xml:id="echoid-s3855" xml:space="preserve">:) CK. </s>
  <s xml:id="echoid-s3856" xml:space="preserve">AC. <lb/></s>
  <s xml:id="echoid-s3857" xml:space="preserve">quod, è præmonſtratis, reflectioni proprium eſt. </s>
  <s xml:id="echoid-s3858" xml:space="preserve">ergò liquet pro-<lb/>poſitum.</s>
  <s xml:id="echoid-s3859" xml:space="preserve"/>
</p>
<div xml:id="echoid-div103" type="float" level="2" n="5">
<note position="left" xlink:label="note-0082-01" xlink:href="note-0082-01a" xml:space="preserve">Fig. 89.</note>
</div>
<p>
  <s xml:id="echoid-s3860" xml:space="preserve">2. </s>
  <s xml:id="echoid-s3861" xml:space="preserve">ltà quidem in hoc caſu; </s>
  <s xml:id="echoid-s3862" xml:space="preserve">at ſi punctum A ponatur aliàs, ut ſit <lb/>AC &amp;</s>
  <s xml:id="echoid-s3863" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s3864" xml:space="preserve">AN; </s>
  <s xml:id="echoid-s3865" xml:space="preserve">reliquis ſtantibus, Sumendum erit intervallum AN <lb/> = √ : </s>
  <s xml:id="echoid-s3866" xml:space="preserve">ACq + Vq; </s>
  <s xml:id="echoid-s3867" xml:space="preserve">ut ſit ANq - ACq = Vq. </s>
  <s xml:id="echoid-s3868" xml:space="preserve">ut poſthac <lb/>conſtabit, ubi de concavis agemus. </s>
  <s xml:id="echoid-s3869" xml:space="preserve">Aliter hoc idem. </s>
  <s xml:id="echoid-s3870" xml:space="preserve">Fiat 2 CK. <lb/></s>
  <s xml:id="echoid-s3871" xml:space="preserve">C B: </s>
  <s xml:id="echoid-s3872" xml:space="preserve">: </s>
  <s xml:id="echoid-s3873" xml:space="preserve">CB. </s>
  <s xml:id="echoid-s3874" xml:space="preserve">F. </s>
  <s xml:id="echoid-s3875" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3876" xml:space="preserve">2 CA. </s>
  <s xml:id="echoid-s3877" xml:space="preserve">CB : </s>
  <s xml:id="echoid-s3878" xml:space="preserve">: </s>
  <s xml:id="echoid-s3879" xml:space="preserve">CB. </s>
  <s xml:id="echoid-s3880" xml:space="preserve">E. </s>
  <s xml:id="echoid-s3881" xml:space="preserve">ſumatúrque CQ = E <lb/>+ F. </s>
  <s xml:id="echoid-s3882" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3883" xml:space="preserve">du@ta QN ad AC perpendicularis circulum ſecet in N. </s>
  <s xml:id="echoid-s3884" xml:space="preserve"><lb/>connexæ AN, KN altera alterius reflexa erit. </s>
  <s xml:id="echoid-s3885" xml:space="preserve">hoc è ſuprà dictis <lb/>liquidò conſectatur. </s>
  <s xml:id="echoid-s3886" xml:space="preserve">At ſi fuerit AN &amp;</s>
  <s xml:id="echoid-s3887" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s3888" xml:space="preserve">AC; </s>
  <s xml:id="echoid-s3889" xml:space="preserve">tum accipi debet <lb/>CQ = F - E; </s>
  <s xml:id="echoid-s3890" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3891" xml:space="preserve">(reliquis nihil immutatis, utì poſtmodùm appa-<lb/>rebit) factum erit.</s>
  <s xml:id="echoid-s3892" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3893" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s3894" xml:space="preserve">Intra caſus hos _Problema_, ceu videtis, facilè conſtruitur; <lb/></s>
  <s xml:id="echoid-s3895" xml:space="preserve">aſt illos; </s>
  <s xml:id="echoid-s3896" xml:space="preserve">alióſque ſpeciales, ſi qui ſunt, excipiendo, generaliter con-<lb/>ceptum omnino Solidum eſt, &amp; </s>
  <s xml:id="echoid-s3897" xml:space="preserve">certè _δυσ@@νον_; </s>
  <s xml:id="echoid-s3898" xml:space="preserve">vix ut aliud a <lb/>_Geometris_ hactenus attentatum difficilius reperiatur. </s>
  <s xml:id="echoid-s3899" xml:space="preserve">Et primò qui-<lb/>dem per lineam extrui, explicaríque poterit ſibi peculiarem, hoc vel <lb/>adſimili modo deſcribendam.</s>
  <s xml:id="echoid-s3900" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3901" xml:space="preserve">Connexâ CA, ſuper diametrum CA deſcribatur circulus AI C; <lb/></s>
  <s xml:id="echoid-s3902" xml:space="preserve">item ſemidiametro CA deſcribatur alter circulus AH G. </s>
  <s xml:id="echoid-s3903" xml:space="preserve">tum à C <lb/>educantur rectæ quotvis CI circulum AICſecantes punctis I; </s>
  <s xml:id="echoid-s3904" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3905" xml:space="preserve">per <lb/>A, I ductæ rectæ circulum AHGſecent punctis H; </s>
  <s xml:id="echoid-s3906" xml:space="preserve">demum per H, <lb/>&amp; </s>
  <s xml:id="echoid-s3907" xml:space="preserve">X rectæ ducantur ipſas CI decuſſantes punctis N. </s>
  <s xml:id="echoid-s3908" xml:space="preserve">per hujuſmodi <lb/>
<anchor type="note" xlink:label="note-0082-02a" xlink:href="note-0082-02"/>
puncta quævis deſignabilia tranſibit linea, _Problematis_ expoſiti ſo-<lb/>lutioni accommodata. </s>
  <s xml:id="echoid-s3909" xml:space="preserve">Sit enim ejus, ac reflectentis circuli quævis <lb/>interſectio N (qualium certè pro reflectentis circuli magnitudine ſub-<lb/>inde quatuor, aliquando tres, modò binæ tantùm erunt) &amp; </s>
  <s xml:id="echoid-s3910" xml:space="preserve">connecta-<lb/>tur AN. </s>
  <s xml:id="echoid-s3911" xml:space="preserve">Et quoniam angulus CIA in Semicirculo rectus eſt, erit <lb/>recta AH biſecta in I. </s>
  <s xml:id="echoid-s3912" xml:space="preserve">adeóque triangula AN I, HNIſibimet æqua-<lb/>lia prorſus &amp; </s>
  <s xml:id="echoid-s3913" xml:space="preserve">æquiangala erunt; </s>
  <s xml:id="echoid-s3914" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3915" xml:space="preserve">ſpeciatim ang. </s>
  <s xml:id="echoid-s3916" xml:space="preserve">INA = ang. <lb/></s>
  <s xml:id="echoid-s3917" xml:space="preserve">IN X. </s>
  <s xml:id="echoid-s3918" xml:space="preserve">unde patet propoſitum.</s>
  <s xml:id="echoid-s3919" xml:space="preserve"/>
</p>
<div xml:id="echoid-div104" type="float" level="2" n="6">
<note position="left" xlink:label="note-0082-02" xlink:href="note-0082-02a" xml:space="preserve">Fig. 90.</note>
</div>
<p>
  <s xml:id="echoid-s3920" xml:space="preserve">V. </s>
  <s xml:id="echoid-s3921" xml:space="preserve">Verùm quoniam (ut pridem admonitum) hujuſmodi conſtructi-<lb/>ones, etſi longè faciliores iis quæ per vulgò receptas lineas peraguntur, <lb/>&amp; </s>
  <s xml:id="echoid-s3922" xml:space="preserve">_Problematum_ naturam magìs in propatulo collocantes à _Geometris_
<pb o="65" file="0083" n="83" rhead=""/>
nihilominùs gravatim admittuntur; </s>
  <s xml:id="echoid-s3923" xml:space="preserve">iſtâ tantummodò raptim inſinua-<lb/>tâ, ſubnectemus alìam ab illorum guſtu non abhorrentem; </s>
  <s xml:id="echoid-s3924" xml:space="preserve">illam <lb/>nempe (quando ſcilicet haud alia melior, ut varias pertentans analyſes, <lb/>&amp; </s>
  <s xml:id="echoid-s3925" xml:space="preserve">hoc in alia complura _Problemata_ transformans exiſtimari poſſum, <lb/>facilè poſſit excogitari; </s>
  <s xml:id="echoid-s3926" xml:space="preserve">quum &amp; </s>
  <s xml:id="echoid-s3927" xml:space="preserve">operæ meæ ſatìs alioquin exerci-<lb/>tatæ nonnunquam videatur parcendum) quam olim _Alhazenus Arabs_ <lb/>ſcriptis commendavit; </s>
  <s xml:id="echoid-s3928" xml:space="preserve">ab horribili tamen illâ prolixitate ſimul ac <lb/>obſcuritate; </s>
  <s xml:id="echoid-s3929" xml:space="preserve">neque non ab incondita ſermonis barbarie nonnihil re-<lb/>purgatam. </s>
  <s xml:id="echoid-s3930" xml:space="preserve">quorſum hoc præmittimus _Lemmaticum Problema._</s>
  <s xml:id="echoid-s3931" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3932" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s3933" xml:space="preserve">Trianguli DPNangulus ad P rectus ſit; </s>
  <s xml:id="echoid-s3934" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3935" xml:space="preserve">in hujus uno cru-<lb/>
<anchor type="note" xlink:label="note-0083-01a" xlink:href="note-0083-01"/>
re PN adſignetur punctum F; </s>
  <s xml:id="echoid-s3936" xml:space="preserve">per F recta ducenda eſt, quæ reli-<lb/>quum latus DP (protractam nempe) ac hypotenuſam DN ità ſecet, <lb/>ut ab illis intercepta ad ſegmentum hypotenuſæ lateri primò contermi-<lb/>num datam obtineat proportionem R ad S.</s>
  <s xml:id="echoid-s3937" xml:space="preserve"/>
</p>
<div xml:id="echoid-div105" type="float" level="2" n="7">
<note position="right" xlink:label="note-0083-01" xlink:href="note-0083-01a" xml:space="preserve">Fig. 91.</note>
</div>
<p>
  <s xml:id="echoid-s3938" xml:space="preserve">Hoc ità peragatur licet. </s>
  <s xml:id="echoid-s3939" xml:space="preserve">Ducatur FH ad PD parallela. </s>
  <s xml:id="echoid-s3940" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3941" xml:space="preserve">Dia-<lb/>metro HN deſcribatur Circulus HFN(is nempe per F tranſibit, ob <lb/>angulum HFNrectum) tum connectatur DF; </s>
  <s xml:id="echoid-s3942" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3943" xml:space="preserve">fiat angulus <lb/>FHI = ang. </s>
  <s xml:id="echoid-s3944" xml:space="preserve">FD N. </s>
  <s xml:id="echoid-s3945" xml:space="preserve">ſit etiam R. </s>
  <s xml:id="echoid-s3946" xml:space="preserve">S:</s>
  <s xml:id="echoid-s3947" xml:space="preserve">: DF. </s>
  <s xml:id="echoid-s3948" xml:space="preserve">T &amp; </s>
  <s xml:id="echoid-s3949" xml:space="preserve">a puncto I duca-<lb/>tur recta ILKdiametrum HN interſecans ad L, &amp; </s>
  <s xml:id="echoid-s3950" xml:space="preserve">circulo occurrens <lb/>in K, ità quidem ut ſit intercepta LK = T (hoc autem quomodò <lb/>præſtetur in ſuperioribus oſtenſum) denuo per puncta KF trajiciatur <lb/>recta CF, ipſam DP ſecans in X. </s>
  <s xml:id="echoid-s3951" xml:space="preserve">Dico factum, vel ipſe CX. <lb/></s>
  <s xml:id="echoid-s3952" xml:space="preserve">CN:</s>
  <s xml:id="echoid-s3953" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s3954" xml:space="preserve">S. </s>
  <s xml:id="echoid-s3955" xml:space="preserve">connectatur enim recta NK. </s>
  <s xml:id="echoid-s3956" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s3957" xml:space="preserve">quoniam ang FK I <lb/>
<anchor type="note" xlink:label="note-0083-02a" xlink:href="note-0083-02"/>
(vel FH I) = FDN, erit triangulum FDCſimile triangulo <lb/>LK C; </s>
  <s xml:id="echoid-s3958" xml:space="preserve">ac indè FD. </s>
  <s xml:id="echoid-s3959" xml:space="preserve">DC:</s>
  <s xml:id="echoid-s3960" xml:space="preserve">: KL. </s>
  <s xml:id="echoid-s3961" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s3962" xml:space="preserve">item ob ang. </s>
  <s xml:id="echoid-s3963" xml:space="preserve">FKN = ang. <lb/></s>
  <s xml:id="echoid-s3964" xml:space="preserve">FHN = ang. </s>
  <s xml:id="echoid-s3965" xml:space="preserve">XDC; </s>
  <s xml:id="echoid-s3966" xml:space="preserve">erunt triangula XDC, NKCſibi quoque <lb/>ſimilia, proindéque DC. </s>
  <s xml:id="echoid-s3967" xml:space="preserve">CX:</s>
  <s xml:id="echoid-s3968" xml:space="preserve">: CK. </s>
  <s xml:id="echoid-s3969" xml:space="preserve">CN. </s>
  <s xml:id="echoid-s3970" xml:space="preserve">quapropter erit ex <lb/>æquali FD. </s>
  <s xml:id="echoid-s3971" xml:space="preserve">CX :</s>
  <s xml:id="echoid-s3972" xml:space="preserve">: KL. </s>
  <s xml:id="echoid-s3973" xml:space="preserve">CN. </s>
  <s xml:id="echoid-s3974" xml:space="preserve">vel permutando FDCK:</s>
  <s xml:id="echoid-s3975" xml:space="preserve">: CX. </s>
  <s xml:id="echoid-s3976" xml:space="preserve"><lb/>C N; </s>
  <s xml:id="echoid-s3977" xml:space="preserve">hoc eſt FD. </s>
  <s xml:id="echoid-s3978" xml:space="preserve">T (vel R. </s>
  <s xml:id="echoid-s3979" xml:space="preserve">S):</s>
  <s xml:id="echoid-s3980" xml:space="preserve">: CX. </s>
  <s xml:id="echoid-s3981" xml:space="preserve">CN; </s>
  <s xml:id="echoid-s3982" xml:space="preserve">quod faciendum <lb/>erat.</s>
  <s xml:id="echoid-s3983" xml:space="preserve"/>
</p>
<div xml:id="echoid-div106" type="float" level="2" n="8">
<note position="right" xlink:label="note-0083-02" xlink:href="note-0083-02a" xml:space="preserve">_Conſtr. &amp;._</note>
</div>
<p>
  <s xml:id="echoid-s3984" xml:space="preserve">Advertendum eſt autem, quod datum punctum F in recta PN <lb/>indefinitè protenſa variè ſtatui poteſt; </s>
  <s xml:id="echoid-s3985" xml:space="preserve">vel nimirum inter puncta P, <lb/>N; </s>
  <s xml:id="echoid-s3986" xml:space="preserve">vel extra illa partes ad alterutras item quòd in iſtorum caſuum <lb/>ſingulo quoque recta IK (conditione gaudens præſtitutâ) plurifa-<lb/>riàm duci poteſt; </s>
  <s xml:id="echoid-s3987" xml:space="preserve">ut antehac inculcatum; </s>
  <s xml:id="echoid-s3988" xml:space="preserve">unde plures emergent ſolu-<lb/>tiones. </s>
  <s xml:id="echoid-s3989" xml:space="preserve">at quoad omnes caſus perſimilis erit conſtructio, nec ferè di-<lb/>verſa demonſtratio. </s>
  <s xml:id="echoid-s3990" xml:space="preserve">quare cur plura?</s>
  <s xml:id="echoid-s3991" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3992" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s3993" xml:space="preserve">Proponatur jam circulus reflectens (is qui præ oculis, cujus
<pb o="66" file="0084" n="84" rhead=""/>
centrum C) datáque ſint duo puncta A, X; </s>
  <s xml:id="echoid-s3994" xml:space="preserve">reperiendum eſt in <lb/>circumferentia punctum aliquod; </s>
  <s xml:id="echoid-s3995" xml:space="preserve">à quo ductæ ad A, X rectæ, altera ſit <lb/>alterius reflexa. </s>
  <s xml:id="echoid-s3996" xml:space="preserve">‖ Hocità perficimus:</s>
  <s xml:id="echoid-s3997" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s3998" xml:space="preserve">Conjungantur rectæ AC, XC; </s>
  <s xml:id="echoid-s3999" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4000" xml:space="preserve">fiat (ſeorſim) ang. </s>
  <s xml:id="echoid-s4001" xml:space="preserve">δ = {1/2} ang. <lb/></s>
  <s xml:id="echoid-s4002" xml:space="preserve">
<anchor type="note" xlink:label="note-0084-01a" xlink:href="note-0084-01"/>
AC X. </s>
  <s xml:id="echoid-s4003" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4004" xml:space="preserve">in ξ δ crure anguli δ ſumpto liberè puncto π ducatur π V <lb/>ad ξ δ perpendicularis alterum crus ſecans in V; </s>
  <s xml:id="echoid-s4005" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4006" xml:space="preserve">in V π protracta <lb/>capiatur π γ = π V; </s>
  <s xml:id="echoid-s4007" xml:space="preserve">tum dividatur γ V in φ, ut ſit γ φ. </s>
  <s xml:id="echoid-s4008" xml:space="preserve">φ V:</s>
  <s xml:id="echoid-s4009" xml:space="preserve">: XC. </s>
  <s xml:id="echoid-s4010" xml:space="preserve">CA; <lb/></s>
  <s xml:id="echoid-s4011" xml:space="preserve">perque punctum φ trajiciatur κ ξ ſic ut ſit κξ. </s>
  <s xml:id="echoid-s4012" xml:space="preserve">κ v:</s>
  <s xml:id="echoid-s4013" xml:space="preserve">: CX. </s>
  <s xml:id="echoid-s4014" xml:space="preserve">CN. </s>
  <s xml:id="echoid-s4015" xml:space="preserve">denique <lb/>
<anchor type="note" xlink:label="note-0084-02a" xlink:href="note-0084-02"/>
fiat angulus XCNæ qualis angulo ξ κ v; </s>
  <s xml:id="echoid-s4016" xml:space="preserve">erit punctum N quale deſi-<lb/>deramus. </s>
  <s xml:id="echoid-s4017" xml:space="preserve">Nam ducantur XN, ξ v; </s>
  <s xml:id="echoid-s4018" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4019" xml:space="preserve">fiat ang. </s>
  <s xml:id="echoid-s4020" xml:space="preserve">CNG = ang. <lb/></s>
  <s xml:id="echoid-s4021" xml:space="preserve">κ v γ. </s>
  <s xml:id="echoid-s4022" xml:space="preserve">adſumatúrque PG = PN; </s>
  <s xml:id="echoid-s4023" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4024" xml:space="preserve">connectatur XG. </s>
  <s xml:id="echoid-s4025" xml:space="preserve">liquet jam <lb/>trangula XCN, ξ κ v ſimilia fore; </s>
  <s xml:id="echoid-s4026" xml:space="preserve">nec non ipſa CNF, κ v φ; </s>
  <s xml:id="echoid-s4027" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s4028" xml:space="preserve">ipſa XP F, ξ π φ; </s>
  <s xml:id="echoid-s4029" xml:space="preserve">ipsáque demùm XFN, ξ φ v aſſimilari. </s>
  <s xml:id="echoid-s4030" xml:space="preserve">quare <lb/>PF. </s>
  <s xml:id="echoid-s4031" xml:space="preserve">XF:</s>
  <s xml:id="echoid-s4032" xml:space="preserve">: π φ. </s>
  <s xml:id="echoid-s4033" xml:space="preserve">ξ φ. </s>
  <s xml:id="echoid-s4034" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4035" xml:space="preserve">XF. </s>
  <s xml:id="echoid-s4036" xml:space="preserve">FN:</s>
  <s xml:id="echoid-s4037" xml:space="preserve">: ξ φ. </s>
  <s xml:id="echoid-s4038" xml:space="preserve">φ V. </s>
  <s xml:id="echoid-s4039" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4040" xml:space="preserve">ex æquo PF. </s>
  <s xml:id="echoid-s4041" xml:space="preserve"><lb/>FN:</s>
  <s xml:id="echoid-s4042" xml:space="preserve">: π φ. </s>
  <s xml:id="echoid-s4043" xml:space="preserve">φ V. </s>
  <s xml:id="echoid-s4044" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4045" xml:space="preserve">antecedentes duplando 2 PF. </s>
  <s xml:id="echoid-s4046" xml:space="preserve">FN :</s>
  <s xml:id="echoid-s4047" xml:space="preserve">: 2 π φ. </s>
  <s xml:id="echoid-s4048" xml:space="preserve">φ V. </s>
  <s xml:id="echoid-s4049" xml:space="preserve"><lb/>componendóque 2 PF + FN. </s>
  <s xml:id="echoid-s4050" xml:space="preserve">FN:</s>
  <s xml:id="echoid-s4051" xml:space="preserve">: 2 π φ + φ V. </s>
  <s xml:id="echoid-s4052" xml:space="preserve">φ V. </s>
  <s xml:id="echoid-s4053" xml:space="preserve">hoc eſt <lb/>GF. </s>
  <s xml:id="echoid-s4054" xml:space="preserve">FN:</s>
  <s xml:id="echoid-s4055" xml:space="preserve">: γ φ. </s>
  <s xml:id="echoid-s4056" xml:space="preserve">φ V (hoc eſt):</s>
  <s xml:id="echoid-s4057" xml:space="preserve">: XC. </s>
  <s xml:id="echoid-s4058" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s4059" xml:space="preserve">ducatur jam NL ad <lb/>XG parailela; </s>
  <s xml:id="echoid-s4060" xml:space="preserve">quare eſt ang. </s>
  <s xml:id="echoid-s4061" xml:space="preserve">ING = ang. </s>
  <s xml:id="echoid-s4062" xml:space="preserve">G = ang. </s>
  <s xml:id="echoid-s4063" xml:space="preserve">XNG; </s>
  <s xml:id="echoid-s4064" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s4065" xml:space="preserve">XG (XN). </s>
  <s xml:id="echoid-s4066" xml:space="preserve">NL:</s>
  <s xml:id="echoid-s4067" xml:space="preserve">: GF. </s>
  <s xml:id="echoid-s4068" xml:space="preserve">FN:</s>
  <s xml:id="echoid-s4069" xml:space="preserve">: XC. </s>
  <s xml:id="echoid-s4070" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s4071" xml:space="preserve">porro fiàt ang. </s>
  <s xml:id="echoid-s4072" xml:space="preserve"><lb/>LNH = ang. </s>
  <s xml:id="echoid-s4073" xml:space="preserve">XCA; </s>
  <s xml:id="echoid-s4074" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4075" xml:space="preserve">HN protracta ipſi CA occurrat in M; </s>
  <s xml:id="echoid-s4076" xml:space="preserve"><lb/>cſtque proptereà triangulum HNLſimile triangulo HCM; </s>
  <s xml:id="echoid-s4077" xml:space="preserve">idcir-<lb/>cóque HC. </s>
  <s xml:id="echoid-s4078" xml:space="preserve">CM:</s>
  <s xml:id="echoid-s4079" xml:space="preserve">: HN. </s>
  <s xml:id="echoid-s4080" xml:space="preserve">NL. </s>
  <s xml:id="echoid-s4081" xml:space="preserve">ducatur denuò tangens NQ; </s>
  <s xml:id="echoid-s4082" xml:space="preserve">eſt-<lb/>que tum ang. </s>
  <s xml:id="echoid-s4083" xml:space="preserve">PNQ = rect - CNP = rect - κ V π = ang. </s>
  <s xml:id="echoid-s4084" xml:space="preserve"><lb/>δ = {1/2} XC A; </s>
  <s xml:id="echoid-s4085" xml:space="preserve">vel 2 ang. </s>
  <s xml:id="echoid-s4086" xml:space="preserve">PNQ = ang XCA = ang LNH. </s>
  <s xml:id="echoid-s4087" xml:space="preserve"><lb/>verùm erat priùs 2 ang. </s>
  <s xml:id="echoid-s4088" xml:space="preserve">XNF = ang. </s>
  <s xml:id="echoid-s4089" xml:space="preserve">XNL. </s>
  <s xml:id="echoid-s4090" xml:space="preserve">ergo 2 ang XNF <lb/>- 2 ang PNQ = ang XNL- ang. </s>
  <s xml:id="echoid-s4091" xml:space="preserve">LNH. </s>
  <s xml:id="echoid-s4092" xml:space="preserve">hoc eſt 2 ang XNQ <lb/> = ang. </s>
  <s xml:id="echoid-s4093" xml:space="preserve">XNH. </s>
  <s xml:id="echoid-s4094" xml:space="preserve">ergo tangens NQ biſecat angulum XNH; </s>
  <s xml:id="echoid-s4095" xml:space="preserve">indéque <lb/>conſectatur fore rectam HM ipſius XN reflexam; </s>
  <s xml:id="echoid-s4096" xml:space="preserve">ac ideò eſſe XC. </s>
  <s xml:id="echoid-s4097" xml:space="preserve"><lb/>HC:</s>
  <s xml:id="echoid-s4098" xml:space="preserve">: XN. </s>
  <s xml:id="echoid-s4099" xml:space="preserve">HN. </s>
  <s xml:id="echoid-s4100" xml:space="preserve">atqui fuit priùs HC. </s>
  <s xml:id="echoid-s4101" xml:space="preserve">CM:</s>
  <s xml:id="echoid-s4102" xml:space="preserve">: HN. </s>
  <s xml:id="echoid-s4103" xml:space="preserve">NL quare <lb/>jam erit ex æquo XC. </s>
  <s xml:id="echoid-s4104" xml:space="preserve">CM:</s>
  <s xml:id="echoid-s4105" xml:space="preserve">: XN. </s>
  <s xml:id="echoid-s4106" xml:space="preserve">NL (h@c eſt etiam è præmon-<lb/>ſtratis):</s>
  <s xml:id="echoid-s4107" xml:space="preserve">: XC. </s>
  <s xml:id="echoid-s4108" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s4109" xml:space="preserve">unde CM = CA. </s>
  <s xml:id="echoid-s4110" xml:space="preserve">quapropter HM, ipſius <lb/>XN reflexa tranſit per A: </s>
  <s xml:id="echoid-s4111" xml:space="preserve">Quod propoſitum erat efficere.</s>
  <s xml:id="echoid-s4112" xml:space="preserve">‖</s>
</p>
<div xml:id="echoid-div107" type="float" level="2" n="9">
<note position="left" xlink:label="note-0084-01" xlink:href="note-0084-01a" xml:space="preserve">Fig. 92, 93.</note>
<note position="left" xlink:label="note-0084-02" xlink:href="note-0084-02a" xml:space="preserve">L@m. pr@ced.</note>
</div>
<p>
  <s xml:id="echoid-s4113" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s4114" xml:space="preserve">Hujuſce _Problematis_ ità generaliùs propoſiti varii quidem <lb/>caſus ſunt (etenim vel data puncta jacent ambo extra circuh@ re-<lb/>flectentem; </s>
  <s xml:id="echoid-s4115" xml:space="preserve">vel utrumque poſitum eſt intra circulum; </s>
  <s xml:id="echoid-s4116" xml:space="preserve">vel unum intra <lb/>jacet, alterum extra; </s>
  <s xml:id="echoid-s4117" xml:space="preserve">quinetiam in horum caſuum unoquoque pluries <lb/>conficitur negotium) aſt ubique non abſimilis erit conſtructio; </s>
  <s xml:id="echoid-s4118" xml:space="preserve">ſanè <lb/>nimius eſſem; </s>
  <s xml:id="echoid-s4119" xml:space="preserve">meámque pariter ac veſtram patientiam macerarem <lb/>omnes intricati _Problematis_ nodos evolvendo; </s>
  <s xml:id="echoid-s4120" xml:space="preserve">ſuffecerit ejuſce ſpeci-<lb/>men aliquod protuliſſe.</s>
  <s xml:id="echoid-s4121" xml:space="preserve"/>
</p>
<pb o="67" file="0085" n="85" rhead=""/>
<p>
  <s xml:id="echoid-s4122" xml:space="preserve">IX Adnotabimus tantùm quòd ex _Problematis_ hujuſce natura con-<lb/>ftructioneque propoſita ſatìs attendenti conſtabit (utique ſicut in _H@-_ <lb/>_potheſibus_ antehac tractatis uberiùs eſt declaratum) duorum tantùm <lb/>ad eaſdem axis partes incidentium reflexosad unum ſeſe punctum de-<lb/>cuſſare. </s>
  <s xml:id="echoid-s4123" xml:space="preserve">nam aliorum unius (qui ſubinde poteſt dari) vel alterius re-<lb/>flexi per ejuſmodi punctum tranſeuntes ad alteris partibus incidentes<unsure/> <lb/>pertinebunt.</s>
  <s xml:id="echoid-s4124" xml:space="preserve">‖ Ex his quadantenus eluceſcit datis puncti radiantis, <lb/>oculíque poſitione deſignari poteſt linea quævis, in qua dicti puncti <lb/>ſpecies apparebit; </s>
  <s xml:id="echoid-s4125" xml:space="preserve">incumbit proximè punctum in ea præciſum deter-<lb/>minare, ad quo eadem conſiſtit. </s>
  <s xml:id="echoid-s4126" xml:space="preserve">eo ſpectat hoc Theoremation.</s>
  <s xml:id="echoid-s4127" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4128" xml:space="preserve">X. </s>
  <s xml:id="echoid-s4129" xml:space="preserve">Ab eodem quocunque puncto A manantes duo radii AN, AR <lb/>
<anchor type="note" xlink:label="note-0085-01a" xlink:href="note-0085-01"/>
in circuli reflectentis peripheria præter illum arcum NR (qui inci-<lb/>dentiæ punctis interjacent) intercipiant arcum PS; </s>
  <s xml:id="echoid-s4130" xml:space="preserve">eorum verò re-<lb/>flexi intercipiant arcum π σ; </s>
  <s xml:id="echoid-s4131" xml:space="preserve">erit arcus π σ æqualis Summæ vel diffe-<lb/>rentiæ dupli arcûs NR, &amp; </s>
  <s xml:id="echoid-s4132" xml:space="preserve">arcûs PS. </s>
  <s xml:id="echoid-s4133" xml:space="preserve">Nam (1) in prima figura; <lb/></s>
  <s xml:id="echoid-s4134" xml:space="preserve">eſt PS + SR + RN = PN = N π = π σ + σ R - RN; </s>
  <s xml:id="echoid-s4135" xml:space="preserve">er-<lb/>gò, pares hinc indè SR, &amp; </s>
  <s xml:id="echoid-s4136" xml:space="preserve">σ R ſubducendo, erit PS + RN = <lb/>π σ - RN. </s>
  <s xml:id="echoid-s4137" xml:space="preserve">proindéque PS + 2 RN = π σ. </s>
  <s xml:id="echoid-s4138" xml:space="preserve">(2). </s>
  <s xml:id="echoid-s4139" xml:space="preserve">in altera figura; </s>
  <s xml:id="echoid-s4140" xml:space="preserve"><lb/>erit PS + SR - RN = PN = N π = RN + R σ - σ π. </s>
  <s xml:id="echoid-s4141" xml:space="preserve">qua-<lb/>rè rurſus æquales auferendo SR, R σ manebit PS - RN = RN <lb/>- σ π unde tranſponendo erit σ π = 2 RN - PS.</s>
  <s xml:id="echoid-s4142" xml:space="preserve"/>
</p>
<div xml:id="echoid-div108" type="float" level="2" n="10">
<note position="right" xlink:label="note-0085-01" xlink:href="note-0085-01a" xml:space="preserve">Fig. 95, 96.</note>
</div>
<p>
  <s xml:id="echoid-s4143" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s4144" xml:space="preserve">Etiam hoc _Lemmation_ adſcribemus: </s>
  <s xml:id="echoid-s4145" xml:space="preserve">Biſecetur recta NP in E; <lb/></s>
  <s xml:id="echoid-s4146" xml:space="preserve">
<anchor type="note" xlink:label="note-0085-02a" xlink:href="note-0085-02"/>
&amp; </s>
  <s xml:id="echoid-s4147" xml:space="preserve">ubivis ſumatur punctum A; </s>
  <s xml:id="echoid-s4148" xml:space="preserve">erit EA = {PA ±: </s>
  <s xml:id="echoid-s4149" xml:space="preserve">NA.</s>
  <s xml:id="echoid-s4150" xml:space="preserve">/2.</s>
  <s xml:id="echoid-s4151" xml:space="preserve">} Nam <lb/>EA = {P N/2} ±: </s>
  <s xml:id="echoid-s4152" xml:space="preserve">AN = {PN ±: </s>
  <s xml:id="echoid-s4153" xml:space="preserve">2 AN / 2} = {PA ±: </s>
  <s xml:id="echoid-s4154" xml:space="preserve">AN.</s>
  <s xml:id="echoid-s4155" xml:space="preserve">/2}</s>
</p>
<div xml:id="echoid-div109" type="float" level="2" n="11">
<note position="right" xlink:label="note-0085-02" xlink:href="note-0085-02a" xml:space="preserve">Fig. 94.</note>
</div>
<p>
  <s xml:id="echoid-s4156" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s4157" xml:space="preserve">Exhinc, ut propoſitum citiùs attingamus, Suppoſito radios <lb/>
<anchor type="note" xlink:label="note-0085-03a" xlink:href="note-0085-03"/>
A N, AR (quoad caſum præſentem) ſibi quàm proximos incidere, <lb/>punctum deſignabimus ad quod ipſorum reflexi N π, R σ concurrunt; <lb/></s>
  <s xml:id="echoid-s4158" xml:space="preserve">dicimus utique ſi dicti reflexi concurrant ad Z; </s>
  <s xml:id="echoid-s4159" xml:space="preserve">bifectis ſubtenſis NP, <lb/>N π in E, &amp; </s>
  <s xml:id="echoid-s4160" xml:space="preserve">F; </s>
  <s xml:id="echoid-s4161" xml:space="preserve">fore FZ. </s>
  <s xml:id="echoid-s4162" xml:space="preserve">ZN:</s>
  <s xml:id="echoid-s4163" xml:space="preserve">: EA. </s>
  <s xml:id="echoid-s4164" xml:space="preserve">NA. </s>
  <s xml:id="echoid-s4165" xml:space="preserve">‖ Nam quoniam <lb/>arcus NR, PS ex hypotheſi ſunt indefinitè parvi (ſeu minimi) ſe ha-<lb/>bebunt ut ſuæ ſubtenſæ; </s>
  <s xml:id="echoid-s4166" xml:space="preserve">nec non idem de arcubus NR, π σ dici poteſt. </s>
  <s xml:id="echoid-s4167" xml:space="preserve"><lb/>igitur arc. </s>
  <s xml:id="echoid-s4168" xml:space="preserve">PS. </s>
  <s xml:id="echoid-s4169" xml:space="preserve">RN :</s>
  <s xml:id="echoid-s4170" xml:space="preserve">: PS. </s>
  <s xml:id="echoid-s4171" xml:space="preserve">RN:</s>
  <s xml:id="echoid-s4172" xml:space="preserve">: PA. </s>
  <s xml:id="echoid-s4173" xml:space="preserve">RA. </s>
  <s xml:id="echoid-s4174" xml:space="preserve">(hoc eſt ob RA, <lb/>NA nihil, ex eadem hypotheſi, differentes):</s>
  <s xml:id="echoid-s4175" xml:space="preserve">: PA. </s>
  <s xml:id="echoid-s4176" xml:space="preserve">NA. </s>
  <s xml:id="echoid-s4177" xml:space="preserve">ergò, <lb/>bis componendo, erit PS + 2 RN. </s>
  <s xml:id="echoid-s4178" xml:space="preserve">RN:</s>
  <s xml:id="echoid-s4179" xml:space="preserve">: PA + 2 NA. </s>
  <s xml:id="echoid-s4180" xml:space="preserve">NA.</s>
  <s xml:id="echoid-s4181" xml:space="preserve">
<pb o="68" file="0086" n="86" rhead=""/>
hoc eſt σ π. </s>
  <s xml:id="echoid-s4182" xml:space="preserve">RN:</s>
  <s xml:id="echoid-s4183" xml:space="preserve">: PA + 2 NA. </s>
  <s xml:id="echoid-s4184" xml:space="preserve">NA. </s>
  <s xml:id="echoid-s4185" xml:space="preserve">eſt autem arc σ π. </s>
  <s xml:id="echoid-s4186" xml:space="preserve">RN <lb/>:</s>
  <s xml:id="echoid-s4187" xml:space="preserve">: ſubtenſa σ π. </s>
  <s xml:id="echoid-s4188" xml:space="preserve">RN:</s>
  <s xml:id="echoid-s4189" xml:space="preserve">: π ZZ R:</s>
  <s xml:id="echoid-s4190" xml:space="preserve">: π Z. </s>
  <s xml:id="echoid-s4191" xml:space="preserve">ZN. </s>
  <s xml:id="echoid-s4192" xml:space="preserve">ergo π Z. </s>
  <s xml:id="echoid-s4193" xml:space="preserve">ZN:</s>
  <s xml:id="echoid-s4194" xml:space="preserve">: <lb/>PA + 2 NA. </s>
  <s xml:id="echoid-s4195" xml:space="preserve">NA. </s>
  <s xml:id="echoid-s4196" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4197" xml:space="preserve">componendo π N. </s>
  <s xml:id="echoid-s4198" xml:space="preserve">ZN:</s>
  <s xml:id="echoid-s4199" xml:space="preserve">: PA + 3 NA. </s>
  <s xml:id="echoid-s4200" xml:space="preserve">NA <lb/>&amp; </s>
  <s xml:id="echoid-s4201" xml:space="preserve">antecedentes ſubduplando FN. </s>
  <s xml:id="echoid-s4202" xml:space="preserve">ZN:</s>
  <s xml:id="echoid-s4203" xml:space="preserve">: {PA + 3 NA/2}. </s>
  <s xml:id="echoid-s4204" xml:space="preserve">NA. </s>
  <s xml:id="echoid-s4205" xml:space="preserve">de-<lb/>nique dividendo FZ. </s>
  <s xml:id="echoid-s4206" xml:space="preserve">ZN:</s>
  <s xml:id="echoid-s4207" xml:space="preserve">: {PA + NA / 2}. </s>
  <s xml:id="echoid-s4208" xml:space="preserve">NA. </s>
  <s xml:id="echoid-s4209" xml:space="preserve">eſt autem EA = <lb/>{PA + NA/2}. </s>
  <s xml:id="echoid-s4210" xml:space="preserve">ergò tandem eſt FZZN:</s>
  <s xml:id="echoid-s4211" xml:space="preserve">: EA. </s>
  <s xml:id="echoid-s4212" xml:space="preserve">NA: </s>
  <s xml:id="echoid-s4213" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s4214" xml:space="preserve">E. </s>
  <s xml:id="echoid-s4215" xml:space="preserve">D.</s>
  <s xml:id="echoid-s4216" xml:space="preserve"/>
</p>
<div xml:id="echoid-div110" type="float" level="2" n="12">
<note position="right" xlink:label="note-0085-03" xlink:href="note-0085-03a" xml:space="preserve">Fig. 95, 96.</note>
</div>
<p>
  <s xml:id="echoid-s4217" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s4218" xml:space="preserve">Hinc colligitur punctum Z eſſe locum ipſiſſimum, circa quem <lb/>puncti Z imago conſiſtit; </s>
  <s xml:id="echoid-s4219" xml:space="preserve">oculi reſpectu in reflexo GN π conſ@ituti, <lb/>tanquam ad O. </s>
  <s xml:id="echoid-s4220" xml:space="preserve">etenim ſuperiùs nec ſemel argumentis, ut mihi vide-<lb/>
<anchor type="note" xlink:label="note-0086-01a" xlink:href="note-0086-01"/>
tur, admodum luculentis adfirmatum eſt (ut jam ad inſtar regulæ <lb/>legíſve ratum, fixúmque cenſeri queat iſthic imaginem verſari, ubi <lb/>propiorum incidenti principali (hoc eſt ei cujus reflexus oculi centrum <lb/>tranſiens axis Optici vicem ſubit) radiorum reflexi principalem illum <lb/>reflexum interſecant; </s>
  <s xml:id="echoid-s4221" xml:space="preserve">itaque circa Z in hoc caſu verſatur.</s>
  <s xml:id="echoid-s4222" xml:space="preserve"/>
</p>
<div xml:id="echoid-div111" type="float" level="2" n="13">
<note position="left" xlink:label="note-0086-01" xlink:href="note-0086-01a" xml:space="preserve">Fig. 95, 96.</note>
</div>
<p>
  <s xml:id="echoid-s4223" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s4224" xml:space="preserve">Et hoc argumentatione collegi, non illâ quidem incertâ <lb/>vel ambiguâ, ſed nec ad _Geometrici_ rigoris amuſſim præ illa quam in <lb/>præcedentibus uſurpavi (quanquam &amp; </s>
  <s xml:id="echoid-s4225" xml:space="preserve">hæc è cognatis fontibus pro-<lb/>Huxerit) adeò exactâ; </s>
  <s xml:id="echoid-s4226" xml:space="preserve">concisâ tamen, &amp; </s>
  <s xml:id="echoid-s4227" xml:space="preserve">facili, talíque quæ conclu-<lb/>ſionis adſertæ cauſam apprimè detegit. </s>
  <s xml:id="echoid-s4228" xml:space="preserve">Enim verò ſi pleraque cuncta, <lb/>quæ ſe oggerunt huc attinentia, minutatim ac moroſè perſequi vellem, <lb/>immane quantum tædii (commodo veſtro fortaſſè non tanto) mihi-<lb/>met accerſerem, &amp; </s>
  <s xml:id="echoid-s4229" xml:space="preserve">temporis plurimum veſtri pariter ac mei exhauri-<lb/>rem. </s>
  <s xml:id="echoid-s4230" xml:space="preserve">ſuffecerit itaque jam, &amp; </s>
  <s xml:id="echoid-s4231" xml:space="preserve">poſthac in reliquis Hypotheſibus ſuffi-<lb/>ciat, viâ quàm breviſſimâ (modò tamen certiſſimâ) metam attingere. <lb/></s>
  <s xml:id="echoid-s4232" xml:space="preserve">De convexis hactenus; </s>
  <s xml:id="echoid-s4233" xml:space="preserve">ad concava proximè nos conferemus, aliquan-<lb/>to breviùs exponenda.</s>
  <s xml:id="echoid-s4234" xml:space="preserve">‖</s>
</p>
<pb o="69" file="0087" n="87"/>
</div>
<div xml:id="echoid-div113" type="section" level="1" n="19">
<head xml:id="echoid-head22" xml:space="preserve"><emph style="sc">Lect.</emph> X.</head>
<p>
  <s xml:id="echoid-s4235" xml:space="preserve">1. </s>
  <s xml:id="echoid-s4236" xml:space="preserve">IN poſtrema Lectione quod ſpectavimus punctum circuli convexo <lb/>alluxit; </s>
  <s xml:id="echoid-s4237" xml:space="preserve">nunc parte@ concavas irradians aliud, at magìs @ π’ π ω, <lb/>contemplabimur. </s>
  <s xml:id="echoid-s4238" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4239" xml:space="preserve">quidem caſuum præcipuorum diverſitatem im-<lb/>primìs diſtinguemus. </s>
  <s xml:id="echoid-s4240" xml:space="preserve">Nempe radiet punctum A in circulum re-<lb/>flectentem, cujus centrum C; </s>
  <s xml:id="echoid-s4241" xml:space="preserve">connexaque recta AC protendatur in-<lb/>definitè; </s>
  <s xml:id="echoid-s4242" xml:space="preserve">quo poſito.</s>
  <s xml:id="echoid-s4243" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4244" xml:space="preserve">II. </s>
  <s xml:id="echoid-s4245" xml:space="preserve">1. </s>
  <s xml:id="echoid-s4246" xml:space="preserve">Incidat radius AN; </s>
  <s xml:id="echoid-s4247" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4248" xml:space="preserve">ſit AN = AC; </s>
  <s xml:id="echoid-s4249" xml:space="preserve">erit ipſius AN <lb/>reflexus, puta N _a_, ad AC parallelus.</s>
  <s xml:id="echoid-s4250" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 97.</note>
<p>
  <s xml:id="echoid-s4251" xml:space="preserve">Hoc è ſuprà generatim oſtenſis conſtat; </s>
  <s xml:id="echoid-s4252" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4253" xml:space="preserve">facilè jam patet, con-<lb/>nexâ CA. </s>
  <s xml:id="echoid-s4254" xml:space="preserve">etenim eſt ang. </s>
  <s xml:id="echoid-s4255" xml:space="preserve">ACN = AN C; </s>
  <s xml:id="echoid-s4256" xml:space="preserve">ob AC, AN, ex <lb/>Hypotheſi pares; </s>
  <s xml:id="echoid-s4257" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4258" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s4259" xml:space="preserve">ANC = _a_ NC, propter reflectionem; <lb/></s>
  <s xml:id="echoid-s4260" xml:space="preserve">adeóque ang. </s>
  <s xml:id="echoid-s4261" xml:space="preserve">ACN = _a_ NC; </s>
  <s xml:id="echoid-s4262" xml:space="preserve">unde ſunt AC, N _a_ ſibi paralle-<lb/>læ.</s>
  <s xml:id="echoid-s4263" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4264" xml:space="preserve">III. </s>
  <s xml:id="echoid-s4265" xml:space="preserve">2. </s>
  <s xml:id="echoid-s4266" xml:space="preserve">Incidat radius AM major ipsâ AC; </s>
  <s xml:id="echoid-s4267" xml:space="preserve">ejus reflexus (puta <lb/>
<anchor type="note" xlink:label="note-0087-02a" xlink:href="note-0087-02"/>
M _a_) cum axe directè procedens conveniet ultra centrum, reſpectu <lb/>puncti A; </s>
  <s xml:id="echoid-s4268" xml:space="preserve">(hoc eſt centrum C puncto radianti, concurſuique inter-<lb/>jacebit).</s>
  <s xml:id="echoid-s4269" xml:space="preserve"/>
</p>
<div xml:id="echoid-div113" type="float" level="2" n="1">
<note position="right" xlink:label="note-0087-02" xlink:href="note-0087-02a" xml:space="preserve">Fig. 98.</note>
</div>
<p>
  <s xml:id="echoid-s4270" xml:space="preserve">Nam ob AM &amp;</s>
  <s xml:id="echoid-s4271" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4272" xml:space="preserve">AC, erit ang. </s>
  <s xml:id="echoid-s4273" xml:space="preserve">ACM&amp;</s>
  <s xml:id="echoid-s4274" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4275" xml:space="preserve">AMC = CM _a_. <lb/></s>
  <s xml:id="echoid-s4276" xml:space="preserve">ergo ang. </s>
  <s xml:id="echoid-s4277" xml:space="preserve">BCM+ CM _a_ &amp;</s>
  <s xml:id="echoid-s4278" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4279" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s4280" xml:space="preserve">BCM+ ACM = 2 rect. </s>
  <s xml:id="echoid-s4281" xml:space="preserve"><lb/>quare M _a_, CB convenient infra CM ad partes _a_ B; </s>
  <s xml:id="echoid-s4282" xml:space="preserve">velut <lb/>ad K.</s>
  <s xml:id="echoid-s4283" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4284" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s4285" xml:space="preserve">3. </s>
  <s xml:id="echoid-s4286" xml:space="preserve">Incidat radius AR; </s>
  <s xml:id="echoid-s4287" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4288" xml:space="preserve">ſit AR minor ipsâ AC; </s>
  <s xml:id="echoid-s4289" xml:space="preserve">ejus <lb/>
<anchor type="note" xlink:label="note-0087-03a" xlink:href="note-0087-03"/>
reflexus, puta R _a_, axi retrò protractus occurret. </s>
  <s xml:id="echoid-s4290" xml:space="preserve">(hoceſt ut radians <lb/>centro, concurſuíque ſit interjectum).</s>
  <s xml:id="echoid-s4291" xml:space="preserve"/>
</p>
<div xml:id="echoid-div114" type="float" level="2" n="2">
<note position="right" xlink:label="note-0087-03" xlink:href="note-0087-03a" xml:space="preserve">Fig. 99.</note>
</div>
<p>
  <s xml:id="echoid-s4292" xml:space="preserve">Nam hîc ob AR &amp;</s>
  <s xml:id="echoid-s4293" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4294" xml:space="preserve">AC; </s>
  <s xml:id="echoid-s4295" xml:space="preserve">erit ang. </s>
  <s xml:id="echoid-s4296" xml:space="preserve">ACR&amp;</s>
  <s xml:id="echoid-s4297" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4298" xml:space="preserve">ang ARC = ang. <lb/></s>
  <s xml:id="echoid-s4299" xml:space="preserve">_a_ RC. </s>
  <s xml:id="echoid-s4300" xml:space="preserve">quapropter ang. </s>
  <s xml:id="echoid-s4301" xml:space="preserve">DCR+ _a_ RC &amp;</s>
  <s xml:id="echoid-s4302" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4303" xml:space="preserve">2 rect. </s>
  <s xml:id="echoid-s4304" xml:space="preserve">unde patet ipſas <lb/>D C, _a_ R portractas infra CR concurrere.</s>
  <s xml:id="echoid-s4305" xml:space="preserve"/>
</p>
<pb o="70" file="0088" n="88" rhead=""/>
<p>
  <s xml:id="echoid-s4306" xml:space="preserve">V. </s>
  <s xml:id="echoid-s4307" xml:space="preserve">Horum caſuum primus ad unum duntaxat ab una axis parte radi-<lb/>um pertinet, qui reliquos aliis caſibus convenientes medius diſterminat. <lb/></s>
  <s xml:id="echoid-s4308" xml:space="preserve">
<anchor type="note" xlink:label="note-0088-01a" xlink:href="note-0088-01"/>
de poſteribus itaque duobus ſeparatim paullò diſpiciamus, Sit jam <lb/>itaque primò AC = AG = A γ; </s>
  <s xml:id="echoid-s4309" xml:space="preserve">unde quilibet incidens cavo GB γ <lb/>radius (ut AN) major erit quam AC; </s>
  <s xml:id="echoid-s4310" xml:space="preserve">hujus itaque reflexus axem <lb/>ſecet puncto K; </s>
  <s xml:id="echoid-s4311" xml:space="preserve">dico, ſi ſemidiameter CB dividatur in Z; </s>
  <s xml:id="echoid-s4312" xml:space="preserve">ut ſit CZ. <lb/></s>
  <s xml:id="echoid-s4313" xml:space="preserve">ZB:</s>
  <s xml:id="echoid-s4314" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s4315" xml:space="preserve">AB; </s>
  <s xml:id="echoid-s4316" xml:space="preserve">fore CK &amp;</s>
  <s xml:id="echoid-s4317" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4318" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s4319" xml:space="preserve">etenim ob angulum ANK <lb/>
<anchor type="note" xlink:label="note-0088-02a" xlink:href="note-0088-02"/>
biſectum, erit AC. </s>
  <s xml:id="echoid-s4320" xml:space="preserve">CK:</s>
  <s xml:id="echoid-s4321" xml:space="preserve">: AN. </s>
  <s xml:id="echoid-s4322" xml:space="preserve">NK. </s>
  <s xml:id="echoid-s4323" xml:space="preserve">vel permutando AC. </s>
  <s xml:id="echoid-s4324" xml:space="preserve">AN <lb/>:</s>
  <s xml:id="echoid-s4325" xml:space="preserve">: CK. </s>
  <s xml:id="echoid-s4326" xml:space="preserve">NK. </s>
  <s xml:id="echoid-s4327" xml:space="preserve">eſt autem AC. </s>
  <s xml:id="echoid-s4328" xml:space="preserve">AB &amp;</s>
  <s xml:id="echoid-s4329" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4330" xml:space="preserve">AC. </s>
  <s xml:id="echoid-s4331" xml:space="preserve">AN ergo AC. </s>
  <s xml:id="echoid-s4332" xml:space="preserve">AB <lb/>&amp;</s>
  <s xml:id="echoid-s4333" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4334" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s4335" xml:space="preserve">NK &amp;</s>
  <s xml:id="echoid-s4336" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4337" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s4338" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s4339" xml:space="preserve">ergo cùm ſit, ex hypotheſi, CZ. </s>
  <s xml:id="echoid-s4340" xml:space="preserve">ZB <lb/>:</s>
  <s xml:id="echoid-s4341" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s4342" xml:space="preserve">AB; </s>
  <s xml:id="echoid-s4343" xml:space="preserve">erit CZ. </s>
  <s xml:id="echoid-s4344" xml:space="preserve">ZB &amp;</s>
  <s xml:id="echoid-s4345" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4346" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s4347" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s4348" xml:space="preserve">componendóque CB. <lb/></s>
  <s xml:id="echoid-s4349" xml:space="preserve">ZB. </s>
  <s xml:id="echoid-s4350" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s4351" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4352" xml:space="preserve">CB. </s>
  <s xml:id="echoid-s4353" xml:space="preserve">KB. </s>
  <s xml:id="echoid-s4354" xml:space="preserve">unde ZB &amp;</s>
  <s xml:id="echoid-s4355" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4356" xml:space="preserve">KB; </s>
  <s xml:id="echoid-s4357" xml:space="preserve">ſeu CZ &amp;</s>
  <s xml:id="echoid-s4358" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4359" xml:space="preserve">CK; </s>
  <s xml:id="echoid-s4360" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s4361" xml:space="preserve"><lb/>E. </s>
  <s xml:id="echoid-s4362" xml:space="preserve">D.</s>
  <s xml:id="echoid-s4363" xml:space="preserve"/>
</p>
<div xml:id="echoid-div115" type="float" level="2" n="3">
<note position="left" xlink:label="note-0088-01" xlink:href="note-0088-01a" xml:space="preserve">Fig. 101.</note>
<note position="left" xlink:label="note-0088-02" xlink:href="note-0088-02a" xml:space="preserve">Fig. 100.</note>
</div>
<p>
  <s xml:id="echoid-s4364" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s4365" xml:space="preserve">Hinc punctum Z eſt limes infra quem, Verſus centrum, nullus <lb/>reflexus axem interſecat.</s>
  <s xml:id="echoid-s4366" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4367" xml:space="preserve">_Coroll._ </s>
  <s xml:id="echoid-s4368" xml:space="preserve">Hinc ſi puncta Z, ζ ſint limites punctorum A, _a_ (quorum <lb/>A remotius) erit CZ &amp;</s>
  <s xml:id="echoid-s4369" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4370" xml:space="preserve">C ζ.</s>
  <s xml:id="echoid-s4371" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4372" xml:space="preserve">Nam BC. </s>
  <s xml:id="echoid-s4373" xml:space="preserve">AC &amp;</s>
  <s xml:id="echoid-s4374" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4375" xml:space="preserve">BC. </s>
  <s xml:id="echoid-s4376" xml:space="preserve">_a_ C. </s>
  <s xml:id="echoid-s4377" xml:space="preserve">componendóque AB. </s>
  <s xml:id="echoid-s4378" xml:space="preserve">AC &amp;</s>
  <s xml:id="echoid-s4379" xml:space="preserve">lt; <lb/></s>
  <s xml:id="echoid-s4380" xml:space="preserve">_a_ B. </s>
  <s xml:id="echoid-s4381" xml:space="preserve">_a_ C. </s>
  <s xml:id="echoid-s4382" xml:space="preserve">hoc eſt ZB. </s>
  <s xml:id="echoid-s4383" xml:space="preserve">ZC &amp;</s>
  <s xml:id="echoid-s4384" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4385" xml:space="preserve">ζ B. </s>
  <s xml:id="echoid-s4386" xml:space="preserve">ζ C. </s>
  <s xml:id="echoid-s4387" xml:space="preserve">vel compoſitè CB. </s>
  <s xml:id="echoid-s4388" xml:space="preserve">ZC <lb/>&amp;</s>
  <s xml:id="echoid-s4389" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4390" xml:space="preserve">CB. </s>
  <s xml:id="echoid-s4391" xml:space="preserve">ζ C. </s>
  <s xml:id="echoid-s4392" xml:space="preserve">ergò ZC &amp;</s>
  <s xml:id="echoid-s4393" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4394" xml:space="preserve">ζ C.</s>
  <s xml:id="echoid-s4395" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4396" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s4397" xml:space="preserve">Quinetiam erit in hoc caſu; </s>
  <s xml:id="echoid-s4398" xml:space="preserve">ANq - ACq. </s>
  <s xml:id="echoid-s4399" xml:space="preserve">CNq:</s>
  <s xml:id="echoid-s4400" xml:space="preserve">: <lb/>AC. </s>
  <s xml:id="echoid-s4401" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s4402" xml:space="preserve">Nam ducatur KH ad CN parallela, protractæ AN <lb/>occurrens in H; </s>
  <s xml:id="echoid-s4403" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4404" xml:space="preserve">connectatur CP; </s>
  <s xml:id="echoid-s4405" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4406" xml:space="preserve">eodem planè modo quo ſu-<lb/>periùs (in iis quæ circa convexas partes attigimus) oſtendetur fore <lb/>AN x NP. </s>
  <s xml:id="echoid-s4407" xml:space="preserve">CNq:</s>
  <s xml:id="echoid-s4408" xml:space="preserve">: AK. </s>
  <s xml:id="echoid-s4409" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s4410" xml:space="preserve">unde diviſim erit AN x NP -<lb/>CNq. </s>
  <s xml:id="echoid-s4411" xml:space="preserve">CNq:</s>
  <s xml:id="echoid-s4412" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s4413" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s4414" xml:space="preserve">eſt autem AN x NP = ANq -<lb/>AN x AP = ANq -: </s>
  <s xml:id="echoid-s4415" xml:space="preserve">ACq - CNq = ANq - ACq + <lb/>CNq; </s>
  <s xml:id="echoid-s4416" xml:space="preserve">adeóque AN x NP - CNq = ANq - ACq. </s>
  <s xml:id="echoid-s4417" xml:space="preserve">ergò <lb/>demum erit ANq - ACq. </s>
  <s xml:id="echoid-s4418" xml:space="preserve">CNq:</s>
  <s xml:id="echoid-s4419" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s4420" xml:space="preserve">CK: </s>
  <s xml:id="echoid-s4421" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s4422" xml:space="preserve">E. </s>
  <s xml:id="echoid-s4423" xml:space="preserve">D.</s>
  <s xml:id="echoid-s4424" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4425" xml:space="preserve">Notetur; </s>
  <s xml:id="echoid-s4426" xml:space="preserve">ſi fuerit AC minor ſemiſſe ſemidiametri circuli re-<lb/>flectentis, quòd punctum A duos focos habebit ad eaſdem centri par-<lb/>tes, quorum alter ad partes D, alter ad B pertinebit; </s>
  <s xml:id="echoid-s4427" xml:space="preserve">ſin AC major <lb/>fuerit iſtâ Semiſſe, focis qui ad diverſos vertices B, &amp; </s>
  <s xml:id="echoid-s4428" xml:space="preserve">D pertinent, <lb/>centrum C interjacebit.</s>
  <s xml:id="echoid-s4429" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4430" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s4431" xml:space="preserve">Etiam hoc interſeram _Theorema_, præmiſſis conforme: </s>
  <s xml:id="echoid-s4432" xml:space="preserve">Si <lb/>fiat 2 CK. </s>
  <s xml:id="echoid-s4433" xml:space="preserve">CN:</s>
  <s xml:id="echoid-s4434" xml:space="preserve">: CN. </s>
  <s xml:id="echoid-s4435" xml:space="preserve">F; </s>
  <s xml:id="echoid-s4436" xml:space="preserve">itémque 2 CA. </s>
  <s xml:id="echoid-s4437" xml:space="preserve">CN:</s>
  <s xml:id="echoid-s4438" xml:space="preserve">: CN. </s>
  <s xml:id="echoid-s4439" xml:space="preserve">E;</s>
  <s xml:id="echoid-s4440" xml:space="preserve">
<pb o="71" file="0089" n="89" rhead=""/>
&amp; </s>
  <s xml:id="echoid-s4441" xml:space="preserve">demittatur NQ ad AC perpendicularis, erit CQ = F - E. </s>
  <s xml:id="echoid-s4442" xml:space="preserve">‖ <lb/>Nam (ut ſuprà) eft CA. </s>
  <s xml:id="echoid-s4443" xml:space="preserve">CK:</s>
  <s xml:id="echoid-s4444" xml:space="preserve">: F. </s>
  <s xml:id="echoid-s4445" xml:space="preserve">E. </s>
  <s xml:id="echoid-s4446" xml:space="preserve">quare dividendo erit CA <lb/>- CK. </s>
  <s xml:id="echoid-s4447" xml:space="preserve">CK:</s>
  <s xml:id="echoid-s4448" xml:space="preserve">: F - E. </s>
  <s xml:id="echoid-s4449" xml:space="preserve">E. </s>
  <s xml:id="echoid-s4450" xml:space="preserve">Item hîc erit ANq - ACq = <lb/>2 AC x CQ + CNq. </s>
  <s xml:id="echoid-s4451" xml:space="preserve">adeóque 2 AC x CQ + CNq. </s>
  <s xml:id="echoid-s4452" xml:space="preserve">CNq <lb/>:</s>
  <s xml:id="echoid-s4453" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s4454" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s4455" xml:space="preserve">hoc eſt, (ob CNq = 2 AC x E.) </s>
  <s xml:id="echoid-s4456" xml:space="preserve">2 AC x CQ <lb/>+ 2 AC x E. </s>
  <s xml:id="echoid-s4457" xml:space="preserve">2 AC x E:</s>
  <s xml:id="echoid-s4458" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s4459" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s4460" xml:space="preserve">hoc eſt CQ + E. </s>
  <s xml:id="echoid-s4461" xml:space="preserve">E:</s>
  <s xml:id="echoid-s4462" xml:space="preserve">: <lb/>AC. </s>
  <s xml:id="echoid-s4463" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s4464" xml:space="preserve">quare dividendo CQ. </s>
  <s xml:id="echoid-s4465" xml:space="preserve">E:</s>
  <s xml:id="echoid-s4466" xml:space="preserve">: AC - CK. </s>
  <s xml:id="echoid-s4467" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s4468" xml:space="preserve">ergo <lb/>F - E = CQ: </s>
  <s xml:id="echoid-s4469" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s4470" xml:space="preserve">E. </s>
  <s xml:id="echoid-s4471" xml:space="preserve">D.</s>
  <s xml:id="echoid-s4472" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4473" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s4474" xml:space="preserve">porrò, ſi duorum quorumvis radiorum AN, AR reflexi <lb/>NK, RL axem ſecent punctis K, L; </s>
  <s xml:id="echoid-s4475" xml:space="preserve">erit CK. </s>
  <s xml:id="echoid-s4476" xml:space="preserve">CL:</s>
  <s xml:id="echoid-s4477" xml:space="preserve">: ARq -<lb/>ACq. </s>
  <s xml:id="echoid-s4478" xml:space="preserve">ANq - ACq. </s>
  <s xml:id="echoid-s4479" xml:space="preserve">Nam ob CK. </s>
  <s xml:id="echoid-s4480" xml:space="preserve">AC:</s>
  <s xml:id="echoid-s4481" xml:space="preserve">: CNq. </s>
  <s xml:id="echoid-s4482" xml:space="preserve">ANq <lb/>- ACq. </s>
  <s xml:id="echoid-s4483" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4484" xml:space="preserve">AC. </s>
  <s xml:id="echoid-s4485" xml:space="preserve">CL:</s>
  <s xml:id="echoid-s4486" xml:space="preserve">: ARq - ACq. </s>
  <s xml:id="echoid-s4487" xml:space="preserve">CNq. </s>
  <s xml:id="echoid-s4488" xml:space="preserve">erit ex æquo <lb/>perturbatè CK. </s>
  <s xml:id="echoid-s4489" xml:space="preserve">CL:</s>
  <s xml:id="echoid-s4490" xml:space="preserve">: ARq - ACq. </s>
  <s xml:id="echoid-s4491" xml:space="preserve">ANq - ACq.</s>
  <s xml:id="echoid-s4492" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4493" xml:space="preserve">X. </s>
  <s xml:id="echoid-s4494" xml:space="preserve">Hinc ſi radius AR ſit ipſo AN obliquior; </s>
  <s xml:id="echoid-s4495" xml:space="preserve">erit CK &amp;</s>
  <s xml:id="echoid-s4496" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4497" xml:space="preserve">CL. <lb/></s>
  <s xml:id="echoid-s4498" xml:space="preserve">Nam ARq - ACq &amp;</s>
  <s xml:id="echoid-s4499" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4500" xml:space="preserve">ANq - ACq.</s>
  <s xml:id="echoid-s4501" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4502" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s4503" xml:space="preserve">Hinc palàm eſt reflexos NK, RL ſeſe priùs quàm axem <lb/>decuſſare.</s>
  <s xml:id="echoid-s4504" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4505" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s4506" xml:space="preserve">Accipiantur porrò bini pares arcus NR, RX; </s>
  <s xml:id="echoid-s4507" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4508" xml:space="preserve">inciden-<lb/>tium AN, AR, AX reflexi cum axe conveniant punctis K, L, M; <lb/></s>
  <s xml:id="echoid-s4509" xml:space="preserve">dico ſpatium LM, obliquiorum occurſibus interjectum, majus eſſe <lb/>ſpatio LK, quod rectiorum continetur occurſibus. </s>
  <s xml:id="echoid-s4510" xml:space="preserve">Nam è ſuprà <lb/>monſtratis conſtat eſſe ANq + AXq &amp;</s>
  <s xml:id="echoid-s4511" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4512" xml:space="preserve">2 ARq. </s>
  <s xml:id="echoid-s4513" xml:space="preserve">proindeque fo-<lb/>
<anchor type="note" xlink:label="note-0089-01a" xlink:href="note-0089-01"/>
re ANq + AXq - 2 ACq &amp;</s>
  <s xml:id="echoid-s4514" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4515" xml:space="preserve">2 ARq - 2 ACq. </s>
  <s xml:id="echoid-s4516" xml:space="preserve">ac indè <lb/>ANq. </s>
  <s xml:id="echoid-s4517" xml:space="preserve">- ACq. </s>
  <s xml:id="echoid-s4518" xml:space="preserve">ARq - ACq &amp;</s>
  <s xml:id="echoid-s4519" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4520" xml:space="preserve">ARq - ACq. </s>
  <s xml:id="echoid-s4521" xml:space="preserve">AXq -<lb/>ACq. </s>
  <s xml:id="echoid-s4522" xml:space="preserve">hoc eſt CL. </s>
  <s xml:id="echoid-s4523" xml:space="preserve">CK &amp;</s>
  <s xml:id="echoid-s4524" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4525" xml:space="preserve">CM. </s>
  <s xml:id="echoid-s4526" xml:space="preserve">CL. </s>
  <s xml:id="echoid-s4527" xml:space="preserve">vel CM. </s>
  <s xml:id="echoid-s4528" xml:space="preserve">CL &amp;</s>
  <s xml:id="echoid-s4529" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4530" xml:space="preserve">CL. <lb/></s>
  <s xml:id="echoid-s4531" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s4532" xml:space="preserve">quare CM + CK &amp;</s>
  <s xml:id="echoid-s4533" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4534" xml:space="preserve">2 CL. </s>
  <s xml:id="echoid-s4535" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4536" xml:space="preserve">ideò LM &amp;</s>
  <s xml:id="echoid-s4537" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4538" xml:space="preserve">KL.</s>
  <s xml:id="echoid-s4539" xml:space="preserve"/>
</p>
<div xml:id="echoid-div116" type="float" level="2" n="4">
<note position="right" xlink:label="note-0089-01" xlink:href="note-0089-01a" xml:space="preserve">Fig. 102;</note>
</div>
<p>
  <s xml:id="echoid-s4540" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s4541" xml:space="preserve">Hinc rectiùs ingruens lux à reflectione verſus axem conden-<lb/>fatior evadit.</s>
  <s xml:id="echoid-s4542" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4543" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s4544" xml:space="preserve">Quidnì demùm rurſus ex his inferatur, viſibilis A imaginem <lb/>@irca reflexorum metam Z, oculo uſpiam in AZ conſtituto, appa-<lb/>rere?</s>
  <s xml:id="echoid-s4545" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4546" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s4547" xml:space="preserve">Adverſatur ſaltem (id quod experiendo deprehendetur) ocu-<lb/>lo uſpiam in ZB collocato confuſiorem apparentiam objici; </s>
  <s xml:id="echoid-s4548" xml:space="preserve">quippe
<pb o="72" file="0090" n="90" rhead=""/>
cùm eum tunc reflexi convergentes appellant; </s>
  <s xml:id="echoid-s4549" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4550" xml:space="preserve">imago diſtinctior Z <lb/>poſt oculum conſiſtat. </s>
  <s xml:id="echoid-s4551" xml:space="preserve">quin ejuſmodi complures apparentias obſerva-<lb/>bitisipſi ſi lubet, &amp; </s>
  <s xml:id="echoid-s4552" xml:space="preserve">ex his deducetis ‖</s>
</p>
<p>
  <s xml:id="echoid-s4553" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s4554" xml:space="preserve">Prætereà, _dato oculi centro, velut O, quonodo deſignandus_ <lb/>_ſit ipſum pervadens reflexus_ (ceu N π) è ſuprà tractatis aliquatenus <lb/>adparet. </s>
  <s xml:id="echoid-s4555" xml:space="preserve">nec inibi generaliùs expoſitum _Problema_ libet h@c repo-<lb/>
<anchor type="note" xlink:label="note-0090-01a" xlink:href="note-0090-01"/>
nere.</s>
  <s xml:id="echoid-s4556" xml:space="preserve"/>
</p>
<div xml:id="echoid-div117" type="float" level="2" n="5">
<note position="left" xlink:label="note-0090-01" xlink:href="note-0090-01a" xml:space="preserve">Fig. 103.</note>
</div>
<p>
  <s xml:id="echoid-s4557" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s4558" xml:space="preserve">Quinetiam antedicta recenſendo conſtabit, ſi biſecentur ſubtenſa <lb/>PN in E; </s>
  <s xml:id="echoid-s4559" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4560" xml:space="preserve">ſubtenſa N π in F; </s>
  <s xml:id="echoid-s4561" xml:space="preserve">ac fiat FZ. </s>
  <s xml:id="echoid-s4562" xml:space="preserve">ZN:</s>
  <s xml:id="echoid-s4563" xml:space="preserve">: EA. </s>
  <s xml:id="echoid-s4564" xml:space="preserve">NA; <lb/></s>
  <s xml:id="echoid-s4565" xml:space="preserve">radiantis imaginem, visûs O reſpectu, circa punctum Z conſiſtere. </s>
  <s xml:id="echoid-s4566" xml:space="preserve"><lb/>planè ſimilis eſt diſcurſus, quorſum κικζ@‖</s>
</p>
<p>
  <s xml:id="echoid-s4567" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s4568" xml:space="preserve">Supereſt tantùm, ut de poſteriore quem innuebamus caſu <lb/>paucula ſubdamus. </s>
  <s xml:id="echoid-s4569" xml:space="preserve">Eò, ponatur AC = AG = A γ; </s>
  <s xml:id="echoid-s4570" xml:space="preserve">indè quilibet <lb/>incidens cavo GB γ radius ipsâ AC minor erit; </s>
  <s xml:id="echoid-s4571" xml:space="preserve">ſit talis alicujus <lb/>AN reflexus N π; </s>
  <s xml:id="echoid-s4572" xml:space="preserve">qui nempe retro productus cum axe conveniet; <lb/></s>
  <s xml:id="echoid-s4573" xml:space="preserve">puta ad K. </s>
  <s xml:id="echoid-s4574" xml:space="preserve">Etiam hic præcedentibus conformia deprehendentur, <lb/>&amp; </s>
  <s xml:id="echoid-s4575" xml:space="preserve">ſuppari demonſtrabuntur modo; </s>
  <s xml:id="echoid-s4576" xml:space="preserve">qualia ſunt nempe</s>
</p>
<p>
  <s xml:id="echoid-s4577" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s4578" xml:space="preserve">AC. </s>
  <s xml:id="echoid-s4579" xml:space="preserve">AN:</s>
  <s xml:id="echoid-s4580" xml:space="preserve">: CK. </s>
  <s xml:id="echoid-s4581" xml:space="preserve">KN.</s>
  <s xml:id="echoid-s4582" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4583" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s4584" xml:space="preserve">ACq - ANq. </s>
  <s xml:id="echoid-s4585" xml:space="preserve">CN q:</s>
  <s xml:id="echoid-s4586" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s4587" xml:space="preserve">CK.</s>
  <s xml:id="echoid-s4588" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4589" xml:space="preserve">XXI. </s>
  <s xml:id="echoid-s4590" xml:space="preserve">Radii AR ipſo AN obliquioris reflexus cum axe concurrat <lb/>in L; </s>
  <s xml:id="echoid-s4591" xml:space="preserve">erit CK. </s>
  <s xml:id="echoid-s4592" xml:space="preserve">CL:</s>
  <s xml:id="echoid-s4593" xml:space="preserve">: ACq - ARq. </s>
  <s xml:id="echoid-s4594" xml:space="preserve">ACq - ANq. </s>
  <s xml:id="echoid-s4595" xml:space="preserve">ac indè</s>
</p>
<p>
  <s xml:id="echoid-s4596" xml:space="preserve">XXII. </s>
  <s xml:id="echoid-s4597" xml:space="preserve">CK &amp;</s>
  <s xml:id="echoid-s4598" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4599" xml:space="preserve">CL.</s>
  <s xml:id="echoid-s4600" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4601" xml:space="preserve">XXIII. </s>
  <s xml:id="echoid-s4602" xml:space="preserve">Incidentium rectiorum (pares, ut ſuperiùs, arcus in re-<lb/>flectente ſumendo) reflexi concurſus habent a ſe minoribus intervallis <lb/>diſjunctos. </s>
  <s xml:id="echoid-s4603" xml:space="preserve">hæc, inquam, &amp; </s>
  <s xml:id="echoid-s4604" xml:space="preserve">alia quoad reliquos caſus præmonſtra-<lb/>tis conformia, vel agnata perſimili quoque quoad hunc caſum metho-<lb/>do comprobantur. </s>
  <s xml:id="echoid-s4605" xml:space="preserve">quare pluribus tempero; </s>
  <s xml:id="echoid-s4606" xml:space="preserve">ſed enim id quod ubi-<lb/>que præcipuum etiam hîc exertiùs oſtendam; </s>
  <s xml:id="echoid-s4607" xml:space="preserve">præmiſſo tamen hoc, ad <lb/>ſequentia quoque concidenda non inutili, _Lemmatio:_</s>
  <s xml:id="echoid-s4608" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4609" xml:space="preserve">XXIV. </s>
  <s xml:id="echoid-s4610" xml:space="preserve">Detur recta BC; </s>
  <s xml:id="echoid-s4611" xml:space="preserve">in ea protracta deſignandum eſt punctum, <lb/>velut Z; </s>
  <s xml:id="echoid-s4612" xml:space="preserve">ita ut BZ ad CZ datam obtineat rationem, puta I ad R. </s>
  <s xml:id="echoid-s4613" xml:space="preserve">‖ <lb/>
<anchor type="note" xlink:label="note-0090-02a" xlink:href="note-0090-02"/>
Id facilè ſic exequimur.</s>
  <s xml:id="echoid-s4614" xml:space="preserve">‖</s>
</p>
<div xml:id="echoid-div118" type="float" level="2" n="6">
<note position="left" xlink:label="note-0090-02" xlink:href="note-0090-02a" xml:space="preserve">Fig. 104.</note>
</div>
<pb o="73" file="0091" n="91" rhead=""/>
<p>
  <s xml:id="echoid-s4615" xml:space="preserve">1. </s>
  <s xml:id="echoid-s4616" xml:space="preserve">Si fuerit I &amp;</s>
  <s xml:id="echoid-s4617" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s4618" xml:space="preserve">R; </s>
  <s xml:id="echoid-s4619" xml:space="preserve">fiat I-R. </s>
  <s xml:id="echoid-s4620" xml:space="preserve">R:</s>
  <s xml:id="echoid-s4621" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s4622" xml:space="preserve">CZ; </s>
  <s xml:id="echoid-s4623" xml:space="preserve">quare compo-<lb/>
<anchor type="note" xlink:label="note-0091-01a" xlink:href="note-0091-01"/>
nendo erit I. </s>
  <s xml:id="echoid-s4624" xml:space="preserve">R:</s>
  <s xml:id="echoid-s4625" xml:space="preserve">: BZ. </s>
  <s xml:id="echoid-s4626" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s4627" xml:space="preserve">ergò factum.</s>
  <s xml:id="echoid-s4628" xml:space="preserve"/>
</p>
<div xml:id="echoid-div119" type="float" level="2" n="7">
<note position="right" xlink:label="note-0091-01" xlink:href="note-0091-01a" xml:space="preserve">Fig. 105.</note>
</div>
<p>
  <s xml:id="echoid-s4629" xml:space="preserve">2. </s>
  <s xml:id="echoid-s4630" xml:space="preserve">Sin I &amp;</s>
  <s xml:id="echoid-s4631" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4632" xml:space="preserve">R; </s>
  <s xml:id="echoid-s4633" xml:space="preserve">fiat R - I. </s>
  <s xml:id="echoid-s4634" xml:space="preserve">I :</s>
  <s xml:id="echoid-s4635" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s4636" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s4637" xml:space="preserve">ergò rurſus compo-<lb/>nendo R. </s>
  <s xml:id="echoid-s4638" xml:space="preserve">I:</s>
  <s xml:id="echoid-s4639" xml:space="preserve">: CZ. </s>
  <s xml:id="echoid-s4640" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s4641" xml:space="preserve">vel inversè, I. </s>
  <s xml:id="echoid-s4642" xml:space="preserve">R :</s>
  <s xml:id="echoid-s4643" xml:space="preserve">: BZ. </s>
  <s xml:id="echoid-s4644" xml:space="preserve">CZ.</s>
  <s xml:id="echoid-s4645" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4646" xml:space="preserve">XXV. </s>
  <s xml:id="echoid-s4647" xml:space="preserve">Fiat jam CA. </s>
  <s xml:id="echoid-s4648" xml:space="preserve">AB:</s>
  <s xml:id="echoid-s4649" xml:space="preserve">: CZ. </s>
  <s xml:id="echoid-s4650" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s4651" xml:space="preserve">Dico punctum Z eſſe <lb/>metam, citra quam (reſpectu centri C) nullus reflexus axem de-<lb/>cuſſabit; </s>
  <s xml:id="echoid-s4652" xml:space="preserve">hoc eſt præmiſſis inſiſtendo, fore CK &amp;</s>
  <s xml:id="echoid-s4653" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4654" xml:space="preserve">CZ.</s>
  <s xml:id="echoid-s4655" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4656" xml:space="preserve">Nam ducatur NT circulum contingens ad N. </s>
  <s xml:id="echoid-s4657" xml:space="preserve">erit ergo NK. </s>
  <s xml:id="echoid-s4658" xml:space="preserve">NA <lb/>:</s>
  <s xml:id="echoid-s4659" xml:space="preserve">: KT. </s>
  <s xml:id="echoid-s4660" xml:space="preserve">AT &amp;</s>
  <s xml:id="echoid-s4661" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4662" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s4663" xml:space="preserve">AB. </s>
  <s xml:id="echoid-s4664" xml:space="preserve">quare NK. </s>
  <s xml:id="echoid-s4665" xml:space="preserve">NA + AB. </s>
  <s xml:id="echoid-s4666" xml:space="preserve">BZ &amp;</s>
  <s xml:id="echoid-s4667" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4668" xml:space="preserve">BK. <lb/></s>
  <s xml:id="echoid-s4669" xml:space="preserve">AB + AB. </s>
  <s xml:id="echoid-s4670" xml:space="preserve">BZ = BK. </s>
  <s xml:id="echoid-s4671" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s4672" xml:space="preserve">eſt verò NK. </s>
  <s xml:id="echoid-s4673" xml:space="preserve">NA :</s>
  <s xml:id="echoid-s4674" xml:space="preserve">: CK. </s>
  <s xml:id="echoid-s4675" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s4676" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s4677" xml:space="preserve">AB. </s>
  <s xml:id="echoid-s4678" xml:space="preserve">BZ:</s>
  <s xml:id="echoid-s4679" xml:space="preserve">: CA. </s>
  <s xml:id="echoid-s4680" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s4681" xml:space="preserve">ergò CK. </s>
  <s xml:id="echoid-s4682" xml:space="preserve">CA + CA. </s>
  <s xml:id="echoid-s4683" xml:space="preserve">CZ &amp;</s>
  <s xml:id="echoid-s4684" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4685" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s4686" xml:space="preserve"><lb/>BZ. </s>
  <s xml:id="echoid-s4687" xml:space="preserve">hoc eſt CK. </s>
  <s xml:id="echoid-s4688" xml:space="preserve">CZ&amp;</s>
  <s xml:id="echoid-s4689" xml:space="preserve">lt;</s>
  <s xml:id="echoid-s4690" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s4691" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s4692" xml:space="preserve">vel permutando CK. </s>
  <s xml:id="echoid-s4693" xml:space="preserve">BK &amp;</s>
  <s xml:id="echoid-s4694" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4695" xml:space="preserve"><lb/>CZ. </s>
  <s xml:id="echoid-s4696" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s4697" xml:space="preserve">unde dividendo CB. </s>
  <s xml:id="echoid-s4698" xml:space="preserve">BK &amp;</s>
  <s xml:id="echoid-s4699" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4700" xml:space="preserve">CB. </s>
  <s xml:id="echoid-s4701" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s4702" xml:space="preserve">adeóque BK&amp;</s>
  <s xml:id="echoid-s4703" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4704" xml:space="preserve"><lb/>BZ. </s>
  <s xml:id="echoid-s4705" xml:space="preserve">unde liquet propoſitum.</s>
  <s xml:id="echoid-s4706" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4707" xml:space="preserve">XXVI. </s>
  <s xml:id="echoid-s4708" xml:space="preserve">Exhinc (ut in caſibus antè pertractatis) conſectatur ejuſ-<lb/>modi punctum Z eſſe locum ipſiſſimum imaginis punctum A exhiben-<lb/>tis oculo, puta O, in axe CA conſtituto; </s>
  <s xml:id="echoid-s4709" xml:space="preserve">patétque quàm longè paſſim <lb/>ab Opticis; </s>
  <s xml:id="echoid-s4710" xml:space="preserve">nominatim à noviſſimis _Stevino, Hobbio, Fabriog_ in eo <lb/>aſſignando loco aberratur; </s>
  <s xml:id="echoid-s4711" xml:space="preserve">quorum ex ſententia verſatur is ad pun-<lb/>ctum (puta Q) tanto ſemotum à vertice B intervallo, quanto radi-<lb/>ans A ab ipſo B diſtat. </s>
  <s xml:id="echoid-s4712" xml:space="preserve">id quod præterquam quòd nullâ veriſimili rati-<lb/>one nititur (imò rationi prorſus adverſatur; </s>
  <s xml:id="echoid-s4713" xml:space="preserve">cùm nullus omnino <lb/>radius oculum ingrediatur tanquam à puncto Q proveniens) experi-<lb/>entiâ facilimè refutatur. </s>
  <s xml:id="echoid-s4714" xml:space="preserve">Nam ſi tanquam circa punctum A accenſa <lb/>candela ſpeculo cavo GB γ exponatur, oculo velut ad O ſito longè <lb/>majori diſtans intervallo conſpicietur, quàm ipſo BQ, quod ipſam <lb/>AB exæquat. </s>
  <s xml:id="echoid-s4715" xml:space="preserve">quinimò tantillo verſus centrum illum adducendo non <lb/>æquali diſtantiâ, ſed admodum majori videbitur elongari; </s>
  <s xml:id="echoid-s4716" xml:space="preserve">tantâ <lb/>circiter ad ſenſum, probabilémque conjecturam; </s>
  <s xml:id="echoid-s4717" xml:space="preserve">quantam proportio <lb/>requirit à nobis præſtituta. </s>
  <s xml:id="echoid-s4718" xml:space="preserve">quo circà diſcurſus noſter experientiæ <lb/>ſuffragio conſtabilitur.</s>
  <s xml:id="echoid-s4719" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4720" xml:space="preserve">XXVII. </s>
  <s xml:id="echoid-s4721" xml:space="preserve">Quod demum attinet ad locum imaginis reſpectu visûs <lb/>
<anchor type="note" xlink:label="note-0091-02a" xlink:href="note-0091-02"/>
extra radiationis axem poſiti; </s>
  <s xml:id="echoid-s4722" xml:space="preserve">determinatur is eodem ac in caſibus <lb/>antecedaneis modo; </s>
  <s xml:id="echoid-s4723" xml:space="preserve">biſecando ſcilicet ipſas NP, N @ punctis E, F; <lb/></s>
  <s xml:id="echoid-s4724" xml:space="preserve">faciendóque EA. </s>
  <s xml:id="echoid-s4725" xml:space="preserve">AN:</s>
  <s xml:id="echoid-s4726" xml:space="preserve">: FZ. </s>
  <s xml:id="echoid-s4727" xml:space="preserve">ZN. </s>
  <s xml:id="echoid-s4728" xml:space="preserve">Adnotandum ſaltem in recti-<lb/>oribus reflexis imaginem extra circulum conſiſtere; </s>
  <s xml:id="echoid-s4729" xml:space="preserve">ſed in obliquio@i-<lb/>bus intra illum, nempe ſi fuerit AE &amp;</s>
  <s xml:id="echoid-s4730" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4731" xml:space="preserve">AN punctum Z ultra axem
<pb o="74" file="0092" n="92" rhead=""/>
CB exiſtet; </s>
  <s xml:id="echoid-s4732" xml:space="preserve">ſin AE &amp;</s>
  <s xml:id="echoid-s4733" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s4734" xml:space="preserve">AN, punctum Z verſus π exiſtet; </s>
  <s xml:id="echoid-s4735" xml:space="preserve">ſin <lb/>AE = AN; </s>
  <s xml:id="echoid-s4736" xml:space="preserve">concurſus infinitè diſtabit; </s>
  <s xml:id="echoid-s4737" xml:space="preserve">ſeu proximus reflexus ipſi <lb/>N π parallelus erit.</s>
  <s xml:id="echoid-s4738" xml:space="preserve"/>
</p>
<div xml:id="echoid-div120" type="float" level="2" n="8">
<note position="right" xlink:label="note-0091-02" xlink:href="note-0091-02a" xml:space="preserve">Fig. 106.</note>
</div>
<p>
  <s xml:id="echoid-s4739" xml:space="preserve">Not. </s>
  <s xml:id="echoid-s4740" xml:space="preserve">ductâ AQ ad CB perpendiculari, ſi AE = AN; </s>
  <s xml:id="echoid-s4741" xml:space="preserve">erit <lb/>AN = √{AQ q/3}. </s>
  <s xml:id="echoid-s4742" xml:space="preserve">Nam AQ q = AP x AN = 3 AN x AN = 3 AN x AN <lb/> = 3 ANq; </s>
  <s xml:id="echoid-s4743" xml:space="preserve">itaque punctum N, iſtos caſus diſterminans, facilè de-<lb/>
<anchor type="note" xlink:label="note-0092-01a" xlink:href="note-0092-01"/>
ſignatur.</s>
  <s xml:id="echoid-s4744" xml:space="preserve"/>
</p>
<div xml:id="echoid-div121" type="float" level="2" n="9">
<note position="left" xlink:label="note-0092-01" xlink:href="note-0092-01a" xml:space="preserve">Fig. 106.</note>
</div>
<p>
  <s xml:id="echoid-s4745" xml:space="preserve">Rationem ipſi tantillùm attendentes perſpicietis; </s>
  <s xml:id="echoid-s4746" xml:space="preserve">mihi ſanè cunctas <lb/>evolvendo minutias non animi ſatìs, non otii ſuppetit.</s>
  <s xml:id="echoid-s4747" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4748" xml:space="preserve">XXVIII. </s>
  <s xml:id="echoid-s4749" xml:space="preserve">Juvabit his unam, loco forſan opportuniore prætermiſſam, <lb/>obſervatiunculam attexere. </s>
  <s xml:id="echoid-s4750" xml:space="preserve">Si fuerit Z radiantis A lmago, viciſſim erit <lb/>A radiantis Z lmago. </s>
  <s xml:id="echoid-s4751" xml:space="preserve">è dictis quoad ſpeciales caſus facilè cernitur <lb/>hoc conſectari. </s>
  <s xml:id="echoid-s4752" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s4753" xml:space="preserve">hinc generatim verum apparebit ſatìs: </s>
  <s xml:id="echoid-s4754" xml:space="preserve">Si fue-<lb/>rit Z ipſius A imago, tantum unus idcircò ab A manantium inflexus <lb/>per Z tranſibit. </s>
  <s xml:id="echoid-s4755" xml:space="preserve">(hoc imagini proprium eſſe ſæpiùs in decurſu in-<lb/>culcata ſatìs arguunt, ſuperque) quare reciprocè ſolus unus ab Z <lb/>manantium inflexus per A tranlibit (nam ſi duo tales per A tranſire <lb/>dicantur, etiam indè duo per Z tranſibunt, contra hypotheſin) erit <lb/>igitur A ipſius Z imago. </s>
  <s xml:id="echoid-s4756" xml:space="preserve">Merebatur hæc (compendio bene ſerviens, <lb/>&amp; </s>
  <s xml:id="echoid-s4757" xml:space="preserve">caſus inter ſe varios conferentibus affundens lucem) obſervatio gene-<lb/>ralibus intertexi; </s>
  <s xml:id="echoid-s4758" xml:space="preserve">niſi quòd non omnia ſe nobis ſtatim produnt; </s>
  <s xml:id="echoid-s4759" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4760" xml:space="preserve"><lb/>quædam in abſtractione ſumma non ità facilè vèl explicari poſſunt, <lb/>vel comprobari.</s>
  <s xml:id="echoid-s4761" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4762" xml:space="preserve">A Catoptricis jam aliquando manum. </s>
  <s xml:id="echoid-s4763" xml:space="preserve">quæ contentus ità quadante-<lb/>nus promoviſſe, haud diſparia (certè magìs nova, miniméque pro-<lb/>trita) circa refractiones ſphæricas, ſeu circulares, attentabo.</s>
  <s xml:id="echoid-s4764" xml:space="preserve"/>
</p>
<pb o="75" file="0093" n="93" rhead=""/>
<p>
  <s xml:id="echoid-s4765" xml:space="preserve">I. </s>
  <s xml:id="echoid-s4766" xml:space="preserve">_CAtoptricâ circulari defunctus ad Dioptricam promovemur;_ <lb/></s>
  <s xml:id="echoid-s4767" xml:space="preserve">quorſum incidentium quotcunque refractis unâ ſimulo perâ <lb/>delineandis, adeóque reſractionum ſymptomatis organicè pertentan-<lb/>dis modum imprimìs exponemus, præ cæteris, opinor expeditum. </s>
  <s xml:id="echoid-s4768" xml:space="preserve"><lb/>Seorſim ad v γ æqualem diametro (NG) circuli refringentis deſcri-<lb/>batur circulus v π γ. </s>
  <s xml:id="echoid-s4769" xml:space="preserve">item habeat v γ ad S γ rationem illam, quæ re-<lb/>fractiones determinat (illam autem deinceps, ut antehac, conſtanter <lb/>nuncupabo rationem I ad R) &amp; </s>
  <s xml:id="echoid-s4770" xml:space="preserve">ſuper diametro S γ deſcribatur quo-<lb/>que circulus SH γ. </s>
  <s xml:id="echoid-s4771" xml:space="preserve">Incidat jam radius quilibet MN P, cui con-<lb/>
<anchor type="note" xlink:label="note-0093-01a" xlink:href="note-0093-01"/>
veniens deſignandus eſt refractus. </s>
  <s xml:id="echoid-s4772" xml:space="preserve">ut hoc aſlequamur, circulo adpoſi-<lb/>to à V adaptetur v π = NP; </s>
  <s xml:id="echoid-s4773" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4774" xml:space="preserve">centro γ per π deſcriptus circulus <lb/>ſecet circulum SH γ in H; </s>
  <s xml:id="echoid-s4775" xml:space="preserve">connexáque γ Hcirculum v π γ interſecet <lb/>in ξ. </s>
  <s xml:id="echoid-s4776" xml:space="preserve">demùm connexâ v ξ, circulo NPGaccommodetur NX = <lb/>v ξ; </s>
  <s xml:id="echoid-s4777" xml:space="preserve">erit NX ipſius NP refractus. </s>
  <s xml:id="echoid-s4778" xml:space="preserve">Etenim (ductis GP, GX) eſt <lb/>γ H. </s>
  <s xml:id="echoid-s4779" xml:space="preserve">γ ξ :</s>
  <s xml:id="echoid-s4780" xml:space="preserve">: (γ S γ v :</s>
  <s xml:id="echoid-s4781" xml:space="preserve">:) I. </s>
  <s xml:id="echoid-s4782" xml:space="preserve">R. </s>
  <s xml:id="echoid-s4783" xml:space="preserve">hoc eſt γ π. </s>
  <s xml:id="echoid-s4784" xml:space="preserve">γ ξ:</s>
  <s xml:id="echoid-s4785" xml:space="preserve">:I. </s>
  <s xml:id="echoid-s4786" xml:space="preserve">R. </s>
  <s xml:id="echoid-s4787" xml:space="preserve">hoc eſt <lb/>GP. </s>
  <s xml:id="echoid-s4788" xml:space="preserve">GX:</s>
  <s xml:id="echoid-s4789" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s4790" xml:space="preserve">R. </s>
  <s xml:id="echoid-s4791" xml:space="preserve">cùm itaque ſint ipſæ GP, GX recti ſinus angulo-<lb/>rum GN π, GNX(quorum GNPeſt angulus incidentiæ) liquet <lb/>propoſitum.</s>
  <s xml:id="echoid-s4792" xml:space="preserve"/>
</p>
<div xml:id="echoid-div122" type="float" level="2" n="10">
<note position="right" xlink:label="note-0093-01" xlink:href="note-0093-01a" xml:space="preserve">Fig. 107. <lb/>108.</note>
</div>
<p>
  <s xml:id="echoid-s4793" xml:space="preserve">II. </s>
  <s xml:id="echoid-s4794" xml:space="preserve">Ad ipſa _Symptomata_ progrediamur exponenda radiis ad circu-<lb/>lum refractis competentia; </s>
  <s xml:id="echoid-s4795" xml:space="preserve">quorum illa pro more primò pertractabi-<lb/>
<anchor type="note" xlink:label="note-0093-02a" xlink:href="note-0093-02"/>
mus, quæ radianti puncto conveniunt ad infinitam quaſi diſtantiam <lb/>poſito, ſeu parallelos ad ſenſum radios ejaculanti. </s>
  <s xml:id="echoid-s4796" xml:space="preserve">Quocirca per <lb/>circuli refringentis Centrum C punctúmque de longinquo radians <lb/>protendatur recta AC Z; </s>
  <s xml:id="echoid-s4797" xml:space="preserve">tum fiat BZ. </s>
  <s xml:id="echoid-s4798" xml:space="preserve">CZ:</s>
  <s xml:id="echoid-s4799" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s4800" xml:space="preserve">R; </s>
  <s xml:id="echoid-s4801" xml:space="preserve">nec non di-<lb/>vidatur CZ in F, ut ſit FZ. </s>
  <s xml:id="echoid-s4802" xml:space="preserve">FC:</s>
  <s xml:id="echoid-s4803" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s4804" xml:space="preserve">R; </s>
  <s xml:id="echoid-s4805" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4806" xml:space="preserve">centro F per Z deſcri-<lb/>batur circulus EG Z. </s>
  <s xml:id="echoid-s4807" xml:space="preserve">his peractis, accipiatur jam quilibet ad AC <lb/>parallelus MNP(convexis incidens an concavis partibus perinde <lb/>fuerit) dico ſi recta NC (ab incidentiæ nempe puncto per refrin-<lb/>gentis centrum ducta) circulo EGZprotracta occurrat in G; </s>
  <s xml:id="echoid-s4808" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s4809" xml:space="preserve">
<pb o="76" file="0094" n="94" rhead=""/>
in axe capiatur CK = CG; </s>
  <s xml:id="echoid-s4810" xml:space="preserve">connectatúrque recta NK, fore NK <lb/>ipſius MNPrefractum. </s>
  <s xml:id="echoid-s4811" xml:space="preserve">Connectantur enim rectæ FG, BG; </s>
  <s xml:id="echoid-s4812" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4813" xml:space="preserve"><lb/>quoniam eſt BZ. </s>
  <s xml:id="echoid-s4814" xml:space="preserve">CZ:</s>
  <s xml:id="echoid-s4815" xml:space="preserve">: (I.</s>
  <s xml:id="echoid-s4816" xml:space="preserve">R:</s>
  <s xml:id="echoid-s4817" xml:space="preserve">:) FZ. </s>
  <s xml:id="echoid-s4818" xml:space="preserve">FC; </s>
  <s xml:id="echoid-s4819" xml:space="preserve">erit permutando <lb/>BZ. </s>
  <s xml:id="echoid-s4820" xml:space="preserve">FZ:</s>
  <s xml:id="echoid-s4821" xml:space="preserve">: CZ. </s>
  <s xml:id="echoid-s4822" xml:space="preserve">FC. </s>
  <s xml:id="echoid-s4823" xml:space="preserve">dividendóque BF. </s>
  <s xml:id="echoid-s4824" xml:space="preserve">FZ:</s>
  <s xml:id="echoid-s4825" xml:space="preserve">: FZ. <lb/></s>
  <s xml:id="echoid-s4826" xml:space="preserve">
<anchor type="note" xlink:label="note-0094-01a" xlink:href="note-0094-01"/>
FC. </s>
  <s xml:id="echoid-s4827" xml:space="preserve">itaque patet triangula BF G, GFC(latera ſcilicet habentia <lb/>circa communem angulum GFCproportionalia) ſimilia fore. <lb/></s>
  <s xml:id="echoid-s4828" xml:space="preserve">quamobrem erit BG. </s>
  <s xml:id="echoid-s4829" xml:space="preserve">GF:</s>
  <s xml:id="echoid-s4830" xml:space="preserve">: GC. </s>
  <s xml:id="echoid-s4831" xml:space="preserve">CF. </s>
  <s xml:id="echoid-s4832" xml:space="preserve">ſeu permutatim BG. </s>
  <s xml:id="echoid-s4833" xml:space="preserve">GC <lb/>:</s>
  <s xml:id="echoid-s4834" xml:space="preserve">: GF. </s>
  <s xml:id="echoid-s4835" xml:space="preserve">CF. </s>
  <s xml:id="echoid-s4836" xml:space="preserve">hoc eſt BG. </s>
  <s xml:id="echoid-s4837" xml:space="preserve">GC:</s>
  <s xml:id="echoid-s4838" xml:space="preserve">: FZ. </s>
  <s xml:id="echoid-s4839" xml:space="preserve">CF:</s>
  <s xml:id="echoid-s4840" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s4841" xml:space="preserve">R. </s>
  <s xml:id="echoid-s4842" xml:space="preserve">verum in tri-<lb/>angulis BC G, NCKeſt BC = CN, &amp; </s>
  <s xml:id="echoid-s4843" xml:space="preserve">CG = CK; </s>
  <s xml:id="echoid-s4844" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4845" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s4846" xml:space="preserve"><lb/>BCG = NCK; </s>
  <s xml:id="echoid-s4847" xml:space="preserve">adeóque BG. </s>
  <s xml:id="echoid-s4848" xml:space="preserve">GC:</s>
  <s xml:id="echoid-s4849" xml:space="preserve">: NK. </s>
  <s xml:id="echoid-s4850" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s4851" xml:space="preserve">quare erit quo-<lb/>que NK. </s>
  <s xml:id="echoid-s4852" xml:space="preserve">CK:</s>
  <s xml:id="echoid-s4853" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s4854" xml:space="preserve">R. </s>
  <s xml:id="echoid-s4855" xml:space="preserve">ergò, <anchor type="note" xlink:href="" symbol="*"/>ſecundum generatim antehac oſtenſa,
<anchor type="note" xlink:label="note-0094-02a" xlink:href="note-0094-02"/>
liquet NK ipſius MN refractum exiſtere.</s>
  <s xml:id="echoid-s4856" xml:space="preserve"/>
</p>
<div xml:id="echoid-div123" type="float" level="2" n="11">
<note position="right" xlink:label="note-0093-02" xlink:href="note-0093-02a" xml:space="preserve">Fig. 109.</note>
<note position="left" xlink:label="note-0094-01" xlink:href="note-0094-01a" xml:space="preserve">Fig. 109.</note>
<note symbol="*" position="left" xlink:label="note-0094-02" xlink:href="note-0094-02a" xml:space="preserve">_Lect. 3. nu-_ <lb/>_mero. 10._</note>
</div>
<p>
  <s xml:id="echoid-s4857" xml:space="preserve">_Coroll._ </s>
  <s xml:id="echoid-s4858" xml:space="preserve">Adnotetur eſſe triangula BFG, GFCſimilia; </s>
  <s xml:id="echoid-s4859" xml:space="preserve">ac eſſe <lb/>BG. </s>
  <s xml:id="echoid-s4860" xml:space="preserve">CC:</s>
  <s xml:id="echoid-s4861" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s4862" xml:space="preserve">R; </s>
  <s xml:id="echoid-s4863" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4864" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s4865" xml:space="preserve">BG F = GC F; </s>
  <s xml:id="echoid-s4866" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4867" xml:space="preserve">eſſe BF, FG, <lb/>FC {.</s>
  <s xml:id="echoid-s4868" xml:space="preserve">./.</s>
  <s xml:id="echoid-s4869" xml:space="preserve">.}&amp;</s>
  <s xml:id="echoid-s4870" xml:space="preserve">c.</s>
  <s xml:id="echoid-s4871" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4872" xml:space="preserve">III. </s>
  <s xml:id="echoid-s4873" xml:space="preserve">Ex hoc (ſanè pulchro, perutilíque _Theoremate_) cùm particu-<lb/>laris exoritur methodus hujuſmodi quotcunque refractos expeditiſſimè <lb/>ſeu delineandi, ſeu computandi; </s>
  <s xml:id="echoid-s4874" xml:space="preserve">tum ipſorum præcipua _ſymptomata_ <lb/>facilimè diſcernuntur ac demonſtrantur. </s>
  <s xml:id="echoid-s4875" xml:space="preserve">qualia ſunt, quæ in ſubjectis <lb/>exhibentur _Corollariis._</s>
  <s xml:id="echoid-s4876" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4877" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s4878" xml:space="preserve">Patet hinc punctum Z eſſe limitem ultra quem (reſpectu cen-<lb/>tri) nullus axem interſecat refractus; </s>
  <s xml:id="echoid-s4879" xml:space="preserve">ſeu perpendicularis ipſius AB <lb/>(vel ei ſaltem quàm proximè adjacentis radii) refractum ad Z termi-<lb/>
<anchor type="note" xlink:label="note-0094-03a" xlink:href="note-0094-03"/>
nari. </s>
  <s xml:id="echoid-s4880" xml:space="preserve">quia nimirum eſt CZ &amp;</s>
  <s xml:id="echoid-s4881" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4882" xml:space="preserve">CG, vel CR.</s>
  <s xml:id="echoid-s4883" xml:space="preserve"/>
</p>
<div xml:id="echoid-div124" type="float" level="2" n="12">
<note position="left" xlink:label="note-0094-03" xlink:href="note-0094-03a" xml:space="preserve">Fig. 110.</note>
</div>
<p>
  <s xml:id="echoid-s4884" xml:space="preserve">V. </s>
  <s xml:id="echoid-s4885" xml:space="preserve">Conſequitur etiam, ſi duorum incidentium MN, QR (quorum <lb/>QR ſit obliquior) refracti conveniant cum axe punctis K, L, fore <lb/>CK&amp;</s>
  <s xml:id="echoid-s4886" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4887" xml:space="preserve">CL. </s>
  <s xml:id="echoid-s4888" xml:space="preserve">Etenim ſi rectæ NC, RC ad circulum refractarium <lb/>
<anchor type="note" xlink:label="note-0094-04a" xlink:href="note-0094-04"/>
(itâ circulum EGZmeritò ſubinde nominabimus) producantur, ut <lb/>ipſum ſecent punctis G, H; </s>
  <s xml:id="echoid-s4889" xml:space="preserve">liquet eſſe CG &amp;</s>
  <s xml:id="echoid-s4890" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4891" xml:space="preserve">CH; </s>
  <s xml:id="echoid-s4892" xml:space="preserve">adeóque CK <lb/>&amp;</s>
  <s xml:id="echoid-s4893" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s4894" xml:space="preserve">CL. </s>
  <s xml:id="echoid-s4895" xml:space="preserve">Hinc</s>
</p>
<div xml:id="echoid-div125" type="float" level="2" n="13">
<note position="left" xlink:label="note-0094-04" xlink:href="note-0094-04a" xml:space="preserve">Fig. 111.</note>
</div>
<p>
  <s xml:id="echoid-s4896" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s4897" xml:space="preserve">Ad eaſdem partes incidentium refracti ſeſe priùs interſecant <lb/>quàm axem; </s>
  <s xml:id="echoid-s4898" xml:space="preserve">(veluti puta refracti NK, RL ſeſe decuſſant in X.)</s>
  <s xml:id="echoid-s4899" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4900" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s4901" xml:space="preserve">Quinetiam, ſi in primo caſu per centrum C dueatur recta <lb/>
<anchor type="note" xlink:label="note-0094-05a" xlink:href="note-0094-05"/>
VI ad BZ perpendicularis, dictóque circulo refractario occurrens <lb/>ad I; </s>
  <s xml:id="echoid-s4902" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4903" xml:space="preserve">fiat CY = CI, patet punctum Y eſſe limitem refractionis
<pb o="77" file="0095" n="95" rhead=""/>
citeriorem. </s>
  <s xml:id="echoid-s4904" xml:space="preserve">erit enim connexa VY refractus obliquiſſimi radii, ceu <lb/>T V, circulum refringentem contingentis.</s>
  <s xml:id="echoid-s4905" xml:space="preserve"/>
</p>
<div xml:id="echoid-div126" type="float" level="2" n="14">
<note position="left" xlink:label="note-0094-05" xlink:href="note-0094-05a" xml:space="preserve">Fig. 111.</note>
</div>
<p>
  <s xml:id="echoid-s4906" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s4907" xml:space="preserve">Item, in ſecundo caſu ſi recta CVIcirculum EGZtan-<lb/>
<anchor type="note" xlink:label="note-0095-01a" xlink:href="note-0095-01"/>
gat in I, &amp; </s>
  <s xml:id="echoid-s4908" xml:space="preserve">adſumatur CY = CI, erit punctum Y citimus alter <lb/>refractorum limes. </s>
  <s xml:id="echoid-s4909" xml:space="preserve">Etenim connexa VY refractus erit incidentis <lb/>(puta VT) ad BC paralleli; </s>
  <s xml:id="echoid-s4910" xml:space="preserve">qui certè cunctorum obliquiſſimus <lb/>erit hujuſmodi refractionem patientium. </s>
  <s xml:id="echoid-s4911" xml:space="preserve">quum enim (<anchor type="note" xlink:href="" symbol="*"/>è præmiſſis)
<anchor type="note" xlink:label="note-0095-02a" xlink:href="note-0095-02"/>
connexâ FI, ſit FI. </s>
  <s xml:id="echoid-s4912" xml:space="preserve">CF:</s>
  <s xml:id="echoid-s4913" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s4914" xml:space="preserve">R. </s>
  <s xml:id="echoid-s4915" xml:space="preserve">hoc eſt ſinus rectus anguli FCI <lb/>(vel anguli CV T) ad ſinum totum, ut I ad R; </s>
  <s xml:id="echoid-s4916" xml:space="preserve">nullus ipſo TV <lb/>obliquior medium BNVpenetrabit; </s>
  <s xml:id="echoid-s4917" xml:space="preserve">at ipſe quicunque talis reper-<lb/>cutietur; </s>
  <s xml:id="echoid-s4918" xml:space="preserve">velut φ ψ in φ ξ.</s>
  <s xml:id="echoid-s4919" xml:space="preserve"/>
</p>
<div xml:id="echoid-div127" type="float" level="2" n="15">
<note position="right" xlink:label="note-0095-01" xlink:href="note-0095-01a" xml:space="preserve">Fig. 112.</note>
<note symbol="*" position="right" xlink:label="note-0095-02" xlink:href="note-0095-02a" xml:space="preserve">_Lect. 3. num. 7._</note>
</div>
<p>
  <s xml:id="echoid-s4920" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s4921" xml:space="preserve">Cæterùm hìc (tametſi præter ordlnem non nihil, extráque <lb/>ſuum locum) egregiam quandam &amp; </s>
  <s xml:id="echoid-s4922" xml:space="preserve">præſertim notabilem iſtius, quem <lb/>nuncupavimus, refractarii circuli proprietatem interſeremus: </s>
  <s xml:id="echoid-s4923" xml:space="preserve">Om-<lb/>nium à puncto B promanantium, &amp; </s>
  <s xml:id="echoid-s4924" xml:space="preserve">à circuli EGZcavis partibus <lb/>refractionem patentium (juxta caſus prænominatos reſpectivam) re-<lb/>fracti per punctum C tranſibunt.</s>
  <s xml:id="echoid-s4925" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4926" xml:space="preserve">Nam ejuſmodi quilibet incidat radius BG, &amp; </s>
  <s xml:id="echoid-s4927" xml:space="preserve">(ſtantibus quæ præ-<lb/>
<anchor type="note" xlink:label="note-0095-03a" xlink:href="note-0095-03"/>
ſtructa præmonſtratáque ſunt) triangula BG F, GCFſimilia ſunt; <lb/></s>
  <s xml:id="echoid-s4928" xml:space="preserve">angulúſque BGFpar angulo GC F; </s>
  <s xml:id="echoid-s4929" xml:space="preserve">itémque FG. </s>
  <s xml:id="echoid-s4930" xml:space="preserve">CF:</s>
  <s xml:id="echoid-s4931" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s4932" xml:space="preserve">R; </s>
  <s xml:id="echoid-s4933" xml:space="preserve"><lb/>eſt autem FG ad CF, _ut Sinus anguli_ GCF_hoc eſt anguli BGF)_ <lb/>_ad Sinum anguli CG F. </s>
  <s xml:id="echoid-s4934" xml:space="preserve">ergò Sinus anguli_ BGF_(qui eſt angulus_ <lb/>_incidentiæ) ad Sinum anguli_ CGFſe habet, ut I ad R. </s>
  <s xml:id="echoid-s4935" xml:space="preserve">ergò CG β eſt <lb/>refractus ipſius BG: </s>
  <s xml:id="echoid-s4936" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s4937" xml:space="preserve">E. </s>
  <s xml:id="echoid-s4938" xml:space="preserve">D.</s>
  <s xml:id="echoid-s4939" xml:space="preserve"/>
</p>
<div xml:id="echoid-div128" type="float" level="2" n="16">
<note position="right" xlink:label="note-0095-03" xlink:href="note-0095-03a" xml:space="preserve">Fig. 113, <lb/>114.</note>
</div>
<p>
  <s xml:id="echoid-s4940" xml:space="preserve">_Nota._ </s>
  <s xml:id="echoid-s4941" xml:space="preserve">Si qui ad convexas hujuſce circuli partes incidunt, ità re-<lb/>flectantur, ut perpetuo Sinus anguli incidentiæ ad Sinum anguli reflexi <lb/>ſe habeat ut I ad R; </s>
  <s xml:id="echoid-s4942" xml:space="preserve">etiam reflexi per C tranſibunt.</s>
  <s xml:id="echoid-s4943" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4944" xml:space="preserve">Hinc habetur unum (quoad hos caſus) è præcipuis in _Dioptrica_ <lb/>deſideratum, perquam utile; </s>
  <s xml:id="echoid-s4945" xml:space="preserve">Superficies ſimpliciſſima radios ab uno <lb/>puncto procedentes ità refringens, ut tanquam ab altero proveniant; <lb/></s>
  <s xml:id="echoid-s4946" xml:space="preserve">id quod demonſtrationis adductus commoditate _Corollarii_ loco (licèt <lb/>ad aliam pertinens hypotheſin) hic apponere non dubitavi, redeamus <lb/>è diverticulo.</s>
  <s xml:id="echoid-s4947" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4948" xml:space="preserve">X. </s>
  <s xml:id="echoid-s4949" xml:space="preserve">Notandum porrò, quòd diverſos refringentes circulos, iíſque <lb/>competentes, modo præſtituto determinatos, refractarios adſumendo, <lb/>rectæ CB, EZ, CE, CZ, CF eaſdem in uno, quas in altero quovis <lb/>proportiones obſervant; </s>
  <s xml:id="echoid-s4950" xml:space="preserve">id quod facilimè demonſtratur; </s>
  <s xml:id="echoid-s4951" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4952" xml:space="preserve">ſatís elu-
<pb o="78" file="0096" n="96" rhead=""/>
ceſcit ex eo, quod earum omnium ad ſe proportiones in eodem ubique <lb/>modo fundantur in una ratione I ad R. </s>
  <s xml:id="echoid-s4953" xml:space="preserve">verbis, &amp; </s>
  <s xml:id="echoid-s4954" xml:space="preserve">Schematis effin-<lb/>gendis parco. </s>
  <s xml:id="echoid-s4955" xml:space="preserve">Pro ſequentibus hæc adjungo _Lemmatia._</s>
  <s xml:id="echoid-s4956" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4957" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s4958" xml:space="preserve">1. </s>
  <s xml:id="echoid-s4959" xml:space="preserve">Sint tria quanta A, B, C (quorum maximum A) ſe dein-<lb/>ceps æqualiter excedentia; </s>
  <s xml:id="echoid-s4960" xml:space="preserve">ſint etiam altera totidem M, N, O; </s>
  <s xml:id="echoid-s4961" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s4962" xml:space="preserve"><lb/>ſit A. </s>
  <s xml:id="echoid-s4963" xml:space="preserve">B:</s>
  <s xml:id="echoid-s4964" xml:space="preserve">: MN; </s>
  <s xml:id="echoid-s4965" xml:space="preserve">ac B, C:</s>
  <s xml:id="echoid-s4966" xml:space="preserve">: N, O; </s>
  <s xml:id="echoid-s4967" xml:space="preserve">dico fore quoque tria M, N, <lb/>O in ratione continua _Aritbmetica._ </s>
  <s xml:id="echoid-s4968" xml:space="preserve">Nam ob A. </s>
  <s xml:id="echoid-s4969" xml:space="preserve">B:</s>
  <s xml:id="echoid-s4970" xml:space="preserve">: M. </s>
  <s xml:id="echoid-s4971" xml:space="preserve">N. </s>
  <s xml:id="echoid-s4972" xml:space="preserve">erit <lb/>diviſim A-B. </s>
  <s xml:id="echoid-s4973" xml:space="preserve">B:</s>
  <s xml:id="echoid-s4974" xml:space="preserve">: M-N. </s>
  <s xml:id="echoid-s4975" xml:space="preserve">N. </s>
  <s xml:id="echoid-s4976" xml:space="preserve">item ob B. </s>
  <s xml:id="echoid-s4977" xml:space="preserve">C:</s>
  <s xml:id="echoid-s4978" xml:space="preserve">: N. </s>
  <s xml:id="echoid-s4979" xml:space="preserve">O. </s>
  <s xml:id="echoid-s4980" xml:space="preserve">erit per <lb/>rationis converſionem B. </s>
  <s xml:id="echoid-s4981" xml:space="preserve">B-C:</s>
  <s xml:id="echoid-s4982" xml:space="preserve">: N. </s>
  <s xml:id="echoid-s4983" xml:space="preserve">N-O. </s>
  <s xml:id="echoid-s4984" xml:space="preserve">ergò erit ex æquo <lb/>A-B. </s>
  <s xml:id="echoid-s4985" xml:space="preserve">B-C:</s>
  <s xml:id="echoid-s4986" xml:space="preserve">: M-N. </s>
  <s xml:id="echoid-s4987" xml:space="preserve">N-O. </s>
  <s xml:id="echoid-s4988" xml:space="preserve">itaque cùm ſit ex Hypotheſi <lb/>A-B = B-C; </s>
  <s xml:id="echoid-s4989" xml:space="preserve">erit etiam M-N = N-O: </s>
  <s xml:id="echoid-s4990" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s4991" xml:space="preserve">E. </s>
  <s xml:id="echoid-s4992" xml:space="preserve">D.</s>
  <s xml:id="echoid-s4993" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s4994" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s4995" xml:space="preserve">2. </s>
  <s xml:id="echoid-s4996" xml:space="preserve">In circuli quadrante ZQ trium arcuum ZG, ZH, ZI <lb/>Sinus recti F α, F β, F γ æqualiter creſcant (ut nempe ſit αβ = βγ) <lb/>dico fore Gα-Hβ&amp;</s>
  <s xml:id="echoid-s4997" xml:space="preserve">lt;</s>
  <s xml:id="echoid-s4998" xml:space="preserve">Hβ-Iγ.</s>
  <s xml:id="echoid-s4999" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Fig. 115.</note>
<p>
  <s xml:id="echoid-s5000" xml:space="preserve">Nam ducatur ſubtenſa GI ipſam Hβ ſecans, in X; </s>
  <s xml:id="echoid-s5001" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5002" xml:space="preserve">ſint XR, <lb/>IS ad FQ parallelæ patet ipſas GR, XS æquari hoc eſt fore <lb/>Gα-Xβ = Xβ-Iγ; </s>
  <s xml:id="echoid-s5003" xml:space="preserve">unde liquidum eſt eſſe Gα-Hβ&amp;</s>
  <s xml:id="echoid-s5004" xml:space="preserve">lt;</s>
  <s xml:id="echoid-s5005" xml:space="preserve">Hβ <lb/>-Iγ: </s>
  <s xml:id="echoid-s5006" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s5007" xml:space="preserve">E. </s>
  <s xml:id="echoid-s5008" xml:space="preserve">D.</s>
  <s xml:id="echoid-s5009" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5010" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s5011" xml:space="preserve">3. </s>
  <s xml:id="echoid-s5012" xml:space="preserve">Sunto concentrici bini circulorum quadrantes FZ X, <lb/>F ζ ξ; </s>
  <s xml:id="echoid-s5013" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5014" xml:space="preserve">ad FZ parallela ducatur recta quævis LG γ; </s>
  <s xml:id="echoid-s5015" xml:space="preserve">circulos inter-<lb/>ſecans punctis G, γ; </s>
  <s xml:id="echoid-s5016" xml:space="preserve">dico fore FZ - LG &amp;</s>
  <s xml:id="echoid-s5017" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s5018" xml:space="preserve">Fζ-Lγ.</s>
  <s xml:id="echoid-s5019" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Fig. 116.</note>
<p>
  <s xml:id="echoid-s5020" xml:space="preserve">Nam connexa FG circulum ζ γξ producta ſecet in T; </s>
  <s xml:id="echoid-s5021" xml:space="preserve">connectan-<lb/>túrque ſubtenſæ ZG, ζ T (hæc ipſam L γ ſecans in S) Patétque jam <lb/>rectas ZG, ζ T parallelas eſſe; </s>
  <s xml:id="echoid-s5022" xml:space="preserve">adeóque quadrangulum ZGSζ fore <lb/>parallelogrammum; </s>
  <s xml:id="echoid-s5023" xml:space="preserve">unde GS = Z ζ; </s>
  <s xml:id="echoid-s5024" xml:space="preserve">adeóque F ζ - LS = FZ <lb/>- LG. </s>
  <s xml:id="echoid-s5025" xml:space="preserve">ergo F ζ- L γ&amp;</s>
  <s xml:id="echoid-s5026" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5027" xml:space="preserve">FZ - LG: </s>
  <s xml:id="echoid-s5028" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s5029" xml:space="preserve">E. </s>
  <s xml:id="echoid-s5030" xml:space="preserve">D.</s>
  <s xml:id="echoid-s5031" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5032" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s5033" xml:space="preserve">Sint jam tres radii paralleli MN, QR, VX, à ſe diſtantes <lb/>æqualiter (hoc eſt ut ductis N v, Rρ, Xξ ad axem AC perpendicu-<lb/>laribus ſit Xξ-Rρ = Rρ-Nv) &amp; </s>
  <s xml:id="echoid-s5034" xml:space="preserve">ipſorum refracti cum axe <lb/>conveniant punctis K, L, O; </s>
  <s xml:id="echoid-s5035" xml:space="preserve">erit obliquiorum concurſibus interjectum <lb/>
<anchor type="note" xlink:label="note-0096-03a" xlink:href="note-0096-03"/>
ſpatium OL majus ſpatio LK, quod à rectiorum occurſibus conti-<lb/>netur.</s>
  <s xml:id="echoid-s5036" xml:space="preserve"/>
</p>
<div xml:id="echoid-div129" type="float" level="2" n="17">
<note position="left" xlink:label="note-0096-03" xlink:href="note-0096-03a" xml:space="preserve">Fig. 117.</note>
</div>
<p>
  <s xml:id="echoid-s5037" xml:space="preserve">Nam ducantur NC, RC, XC circulo refractario occurrentes <lb/>punctis G, H, I; </s>
  <s xml:id="echoid-s5038" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5039" xml:space="preserve">ad has à refractarii centro F ducantur perpendi-<lb/>culares F α, F β, F γ; </s>
  <s xml:id="echoid-s5040" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5041" xml:space="preserve">quoniam triangula CXξ, CFγ ſimilia <lb/>ſunt; </s>
  <s xml:id="echoid-s5042" xml:space="preserve">erit Xξ. </s>
  <s xml:id="echoid-s5043" xml:space="preserve">CX:</s>
  <s xml:id="echoid-s5044" xml:space="preserve">: Fγ. </s>
  <s xml:id="echoid-s5045" xml:space="preserve">CF. </s>
  <s xml:id="echoid-s5046" xml:space="preserve">item ſimili de cauſa, eſt CR (CX).</s>
  <s xml:id="echoid-s5047" xml:space="preserve">
<pb o="79" file="0097" n="97" rhead=""/>
Rρ :</s>
  <s xml:id="echoid-s5048" xml:space="preserve">: CF. </s>
  <s xml:id="echoid-s5049" xml:space="preserve">Fβ quapropter erit ex æquo Xξ. </s>
  <s xml:id="echoid-s5050" xml:space="preserve">Rρ :</s>
  <s xml:id="echoid-s5051" xml:space="preserve">: Fγ. </s>
  <s xml:id="echoid-s5052" xml:space="preserve">Fβ. <lb/></s>
  <s xml:id="echoid-s5053" xml:space="preserve">non diſpare ratione conſtabit eſſe Rρ. </s>
  <s xml:id="echoid-s5054" xml:space="preserve">Nv:</s>
  <s xml:id="echoid-s5055" xml:space="preserve">: Fβ. </s>
  <s xml:id="echoid-s5056" xml:space="preserve">Fα. </s>
  <s xml:id="echoid-s5057" xml:space="preserve">ergò cùm <lb/>tres Xξ, Rρ, NV ſe æqualiter excedant; </s>
  <s xml:id="echoid-s5058" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/>etiam tres Fγ, Fβ, Fα ſe
<anchor type="note" xlink:label="note-0097-01a" xlink:href="note-0097-01"/>
æqualiter excedent; </s>
  <s xml:id="echoid-s5059" xml:space="preserve">unde conſequetur eſſe Cα-Cβ&amp;</s>
  <s xml:id="echoid-s5060" xml:space="preserve">lt;</s>
  <s xml:id="echoid-s5061" xml:space="preserve">Cβ-Cγ; <lb/></s>
  <s xml:id="echoid-s5062" xml:space="preserve">nec non eſſe <anchor type="note" xlink:href="" symbol="*"/> α G-β H&amp;</s>
  <s xml:id="echoid-s5063" xml:space="preserve">lt;</s>
  <s xml:id="echoid-s5064" xml:space="preserve">βH-γI; </s>
  <s xml:id="echoid-s5065" xml:space="preserve">adeóque conjunctim CG
<anchor type="note" xlink:label="note-0097-02a" xlink:href="note-0097-02"/>
- CH &amp;</s>
  <s xml:id="echoid-s5066" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5067" xml:space="preserve">CH - CI; </s>
  <s xml:id="echoid-s5068" xml:space="preserve">hoc eſt CK-CL &amp;</s>
  <s xml:id="echoid-s5069" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5070" xml:space="preserve">CL - CO; </s>
  <s xml:id="echoid-s5071" xml:space="preserve">hoc <lb/>eſt denuò LK &amp;</s>
  <s xml:id="echoid-s5072" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5073" xml:space="preserve">OL: </s>
  <s xml:id="echoid-s5074" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s5075" xml:space="preserve">E. </s>
  <s xml:id="echoid-s5076" xml:space="preserve">D.</s>
  <s xml:id="echoid-s5077" xml:space="preserve"/>
</p>
<div xml:id="echoid-div130" type="float" level="2" n="18">
<note symbol="*" position="right" xlink:label="note-0097-01" xlink:href="note-0097-01a" xml:space="preserve">_21 hujus Lect._ <lb/>_Hy<unsure/>p._</note>
<note symbol="*" position="right" xlink:label="note-0097-02" xlink:href="note-0097-02a" xml:space="preserve">_22 buius Lect_</note>
</div>
<p>
  <s xml:id="echoid-s5078" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s5079" xml:space="preserve">Hinc apparet rectiùs illapſam refringenti lucem magìs inſpiſſa-<lb/>ri; </s>
  <s xml:id="echoid-s5080" xml:space="preserve">versúſque punctum Z in arctius redigi; </s>
  <s xml:id="echoid-s5081" xml:space="preserve">maximam proinde vim <lb/>ejus iſthic exeri; </s>
  <s xml:id="echoid-s5082" xml:space="preserve">focúmque combuſtionis (ad ſolem) ibi verſari.</s>
  <s xml:id="echoid-s5083" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5084" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s5085" xml:space="preserve">Conſectatur etiam radios (hujuſmodi ſaltem parallelos) quò <lb/>rectiores oculo (cujus nempe ſuperficies refractionis munus obeuntes <lb/>aut Sphæricæ ſunt, aut Sphæricas aliquatenus referunt) incidunt, eò <lb/>facilius ab ipſo readunari, ſeu propiùs recolligi.</s>
  <s xml:id="echoid-s5086" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5087" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s5088" xml:space="preserve">Quinimò tandem ex his colligitur viſibilis longinqui puncti <lb/>ſpeciem oculo, in axe poſito, circa punctum Z apparere. </s>
  <s xml:id="echoid-s5089" xml:space="preserve">Etenim ab <lb/>ei adjacentibus partibus refracti cùm præ cæteris perpendiculares (vi <lb/>proinde fortiores, &amp; </s>
  <s xml:id="echoid-s5090" xml:space="preserve">recollectu paratiores) neque non copioſiores <lb/>affluunt; </s>
  <s xml:id="echoid-s5091" xml:space="preserve">quibus ex cauſis imaginis poſitio dependet; </s>
  <s xml:id="echoid-s5092" xml:space="preserve">ut jam ſæpiùs <lb/>admonitum: </s>
  <s xml:id="echoid-s5093" xml:space="preserve">- ἐχ{θρ}όν δέ {μο}ι {ἐστι}ὶν Αὖ<unsure/>πς ἀeιζήλως {ηῤ}ρημένα μυ{θσ}λο {γρ}ύ{ει}ν <lb/>cæterùm hâc defunctus curâ tantiſper reſpirabo.</s>
  <s xml:id="echoid-s5094" xml:space="preserve">‖</s>
</p>
<pb o="80" file="0098" n="98" rhead=""/>
<p>
  <s xml:id="echoid-s5095" xml:space="preserve">I. </s>
  <s xml:id="echoid-s5096" xml:space="preserve">P_Arallelorum ad circulum refractionem patientium in contem-_ <lb/>_platione defixus, præter alia præcipua ſymptomata, locum ultimè_ <lb/>_determinavi, quam iſti repræſentant, imaginis, oculo in axe conſtitu-_ <lb/>_to._ </s>
  <s xml:id="echoid-s5097" xml:space="preserve">res jam poſtulat ut eandem deſiniamus oculi gratiâ ſecùs colloca-<lb/>ti. </s>
  <s xml:id="echoid-s5098" xml:space="preserve">veruntamen unam priùs haud inutilem adnectam obſervationem, ad <lb/>præcedentia ſpectantem; </s>
  <s xml:id="echoid-s5099" xml:space="preserve">hanc utique:</s>
  <s xml:id="echoid-s5100" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5101" xml:space="preserve">II. </s>
  <s xml:id="echoid-s5102" xml:space="preserve">Si duo Segmenta NB R, v βρ latitudines (vel ſubtenſas) NR, <lb/>
<anchor type="note" xlink:label="note-0098-01a" xlink:href="note-0098-01"/>
V ρ æquales habeant; </s>
  <s xml:id="echoid-s5103" xml:space="preserve">quorum V β ρ ad majorem pertineat circulum; <lb/></s>
  <s xml:id="echoid-s5104" xml:space="preserve">hoc cùm potentiùs aduret, tum objectum viſibile clariùs atque di-<lb/>ftinctiùs exhibebit. </s>
  <s xml:id="echoid-s5105" xml:space="preserve">Sint enim C, κ circulorum refringentium cen-<lb/>tra; </s>
  <s xml:id="echoid-s5106" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5107" xml:space="preserve">circuli iis competentes refractarii ſint EG Z, ε γ ζ; </s>
  <s xml:id="echoid-s5108" xml:space="preserve">horúm-<lb/>que centra F; </s>
  <s xml:id="echoid-s5109" xml:space="preserve">φ; </s>
  <s xml:id="echoid-s5110" xml:space="preserve">tum parallelorum punctis N, v incidentium ſint <lb/>refracti ND, V δ dico tum fore D Z&amp;</s>
  <s xml:id="echoid-s5111" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5112" xml:space="preserve">ζ δ. </s>
  <s xml:id="echoid-s5113" xml:space="preserve">‖ Ducantur enim <lb/>rectæ NC G, V χ γ; </s>
  <s xml:id="echoid-s5114" xml:space="preserve">híſque perpendiculares rectæ FL, φ λ. </s>
  <s xml:id="echoid-s5115" xml:space="preserve">éſtque <lb/>C N. </s>
  <s xml:id="echoid-s5116" xml:space="preserve">v π:</s>
  <s xml:id="echoid-s5117" xml:space="preserve">: CN. </s>
  <s xml:id="echoid-s5118" xml:space="preserve">NP:</s>
  <s xml:id="echoid-s5119" xml:space="preserve">: CF. </s>
  <s xml:id="echoid-s5120" xml:space="preserve">FL. </s>
  <s xml:id="echoid-s5121" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5122" xml:space="preserve">v π. </s>
  <s xml:id="echoid-s5123" xml:space="preserve">χ v:</s>
  <s xml:id="echoid-s5124" xml:space="preserve">: φ λ. </s>
  <s xml:id="echoid-s5125" xml:space="preserve">χ φ. </s>
  <s xml:id="echoid-s5126" xml:space="preserve">ergò <lb/>(rationes ſibi pares adjungendo) eſt CN. </s>
  <s xml:id="echoid-s5127" xml:space="preserve">v π + v π. </s>
  <s xml:id="echoid-s5128" xml:space="preserve">κ v:</s>
  <s xml:id="echoid-s5129" xml:space="preserve">: CF. </s>
  <s xml:id="echoid-s5130" xml:space="preserve"><lb/>FL + φ λ. </s>
  <s xml:id="echoid-s5131" xml:space="preserve">nφ. </s>
  <s xml:id="echoid-s5132" xml:space="preserve">hoc eſt CN. </s>
  <s xml:id="echoid-s5133" xml:space="preserve">χ v :</s>
  <s xml:id="echoid-s5134" xml:space="preserve">: CF x φ λ. </s>
  <s xml:id="echoid-s5135" xml:space="preserve">FL x n φ. </s>
  <s xml:id="echoid-s5136" xml:space="preserve">eſt autem <lb/>C N. </s>
  <s xml:id="echoid-s5137" xml:space="preserve">κ v:</s>
  <s xml:id="echoid-s5138" xml:space="preserve">: CF. </s>
  <s xml:id="echoid-s5139" xml:space="preserve">κ φ :</s>
  <s xml:id="echoid-s5140" xml:space="preserve">: CF x φ λ. </s>
  <s xml:id="echoid-s5141" xml:space="preserve">n φ x φ λ. </s>
  <s xml:id="echoid-s5142" xml:space="preserve">quapropter erit CF <lb/>
<anchor type="note" xlink:label="note-0098-02a" xlink:href="note-0098-02"/>
x φ λ. </s>
  <s xml:id="echoid-s5143" xml:space="preserve">FL x κ φ :</s>
  <s xml:id="echoid-s5144" xml:space="preserve">: CF x φ λ. </s>
  <s xml:id="echoid-s5145" xml:space="preserve">n φ x φ λ; </s>
  <s xml:id="echoid-s5146" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5147" xml:space="preserve">idcircò FL x κ φ = κ φ <lb/>x φ λ; </s>
  <s xml:id="echoid-s5148" xml:space="preserve">indeque FL = φ λ. </s>
  <s xml:id="echoid-s5149" xml:space="preserve">hinc conſequetur fore CF-CL&amp;</s>
  <s xml:id="echoid-s5150" xml:space="preserve">gt; <lb/></s>
  <s xml:id="echoid-s5151" xml:space="preserve">κ φ- κ λ nec non FZ - LG&amp;</s>
  <s xml:id="echoid-s5152" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5153" xml:space="preserve">φζ-λγ; </s>
  <s xml:id="echoid-s5154" xml:space="preserve">proindéque con-<lb/>junctim CZ-CG&amp;</s>
  <s xml:id="echoid-s5155" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5156" xml:space="preserve">κ ζ-κ γ; </s>
  <s xml:id="echoid-s5157" xml:space="preserve">hoc eſt CZ-CD&amp;</s>
  <s xml:id="echoid-s5158" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s5159" xml:space="preserve">κ ζ-κ δ; </s>
  <s xml:id="echoid-s5160" xml:space="preserve"><lb/>hoc eſt demum DZ &amp;</s>
  <s xml:id="echoid-s5161" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s5162" xml:space="preserve">δ ζ; </s>
  <s xml:id="echoid-s5163" xml:space="preserve">exhinc lux ab arcu y ζ ρ magìs conſtipata, <lb/>(in ſpatium quippe reſtrictius δ ζ coacta) viol@ntiùs operabitur; </s>
  <s xml:id="echoid-s5164" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5165" xml:space="preserve"><lb/>a fonte magìs ad punctum accedente promanare viſa punctum radians <lb/>diſtinctiùs exhibebit; </s>
  <s xml:id="echoid-s5166" xml:space="preserve">id quod inſtitutum fuit oſtendere; </s>
  <s xml:id="echoid-s5167" xml:space="preserve">quo rei paſſim <lb/>obſervatæ, nec exilis in perſpiciliorum conſtructione usûs ratio con-<lb/>ſtaret. </s>
  <s xml:id="echoid-s5168" xml:space="preserve">‖ In ordinem jam recidimus; </s>
  <s xml:id="echoid-s5169" xml:space="preserve">ut puncti nempe longinqui <lb/>locum apparentem indagemus, oculi reſpectu quomodocunque ſiti-<lb/>quem in finem conſiciendum venit imprimìs hujuſmodi _Problema, re-_ <lb/>ctum definiens in qua locus iſte verſatur:</s>
  <s xml:id="echoid-s5170" xml:space="preserve"/>
</p>
<div xml:id="echoid-div131" type="float" level="2" n="19">
<note position="left" xlink:label="note-0098-01" xlink:href="note-0098-01a" xml:space="preserve">Fig. 118, <lb/>119.</note>
<note position="left" xlink:label="note-0098-02" xlink:href="note-0098-02a" xml:space="preserve">23. 11 Lect.</note>
</div>
<pb o="81" file="0099" n="99" rhead=""/>
<p>
  <s xml:id="echoid-s5171" xml:space="preserve">III. </s>
  <s xml:id="echoid-s5172" xml:space="preserve">Dato circulo refringente; </s>
  <s xml:id="echoid-s5173" xml:space="preserve">punctóque quovis X; </s>
  <s xml:id="echoid-s5174" xml:space="preserve">per punctum <lb/>X ducatur recta, quæ ſit incidentis ad datam poſitione rectam CB <lb/>parallelè refractus.</s>
  <s xml:id="echoid-s5175" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5176" xml:space="preserve">Si punctum datum X ponatur in axe CB; </s>
  <s xml:id="echoid-s5177" xml:space="preserve">facillimè perficitur ne-<lb/>
<anchor type="note" xlink:label="note-0099-01a" xlink:href="note-0099-01"/>
gotium; </s>
  <s xml:id="echoid-s5178" xml:space="preserve">etenim ſi fiat R. </s>
  <s xml:id="echoid-s5179" xml:space="preserve">I:</s>
  <s xml:id="echoid-s5180" xml:space="preserve">: CX. </s>
  <s xml:id="echoid-s5181" xml:space="preserve">T; </s>
  <s xml:id="echoid-s5182" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5183" xml:space="preserve">centro X intervallo ip-<lb/>ſam T adæquante deſcribatur circulus refringentem interſecans in N; <lb/></s>
  <s xml:id="echoid-s5184" xml:space="preserve">è præmiſſis admodum patet connexum NK per N incidentis ad BC <lb/>paralleli refractum eſſe; </s>
  <s xml:id="echoid-s5185" xml:space="preserve">quia ſcilicet eſt. </s>
  <s xml:id="echoid-s5186" xml:space="preserve">CX. </s>
  <s xml:id="echoid-s5187" xml:space="preserve">XN:</s>
  <s xml:id="echoid-s5188" xml:space="preserve">: CX. </s>
  <s xml:id="echoid-s5189" xml:space="preserve">T:</s>
  <s xml:id="echoid-s5190" xml:space="preserve">: <lb/>R. </s>
  <s xml:id="echoid-s5191" xml:space="preserve">I.</s>
  <s xml:id="echoid-s5192" xml:space="preserve"/>
</p>
<div xml:id="echoid-div132" type="float" level="2" n="20">
<note position="right" xlink:label="note-0099-01" xlink:href="note-0099-01a" xml:space="preserve">Fig. 120.</note>
</div>
<p>
  <s xml:id="echoid-s5193" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s5194" xml:space="preserve">Verùm extra caſum hunc, &amp; </s>
  <s xml:id="echoid-s5195" xml:space="preserve">alios particulares nil huc atti-<lb/>nentes, generatim conceptum _Problema_ Solidum eſt, aut pluſquam <lb/>Solidum (ut ex analyſi non difficilè perſpiciatur) &amp; </s>
  <s xml:id="echoid-s5196" xml:space="preserve">certè viâ con-<lb/>ſuetâ, per lineas vulgò receptas, conſtructu perquam arduum &amp; </s>
  <s xml:id="echoid-s5197" xml:space="preserve"><lb/>operoſum; </s>
  <s xml:id="echoid-s5198" xml:space="preserve">ità quidem ut licèt mihi non penitus incomperta ſit metho-<lb/>dus ejuſmodi conſtructionem non unam moliendi, ægrè poſſim addu-<lb/>ci, tantum ut ei temporis, tantum laboris impendam, quantum <lb/>expoſcit; </s>
  <s xml:id="echoid-s5199" xml:space="preserve">ſuffecerit itaque modum indigitare, quo per lineam quan-<lb/>dam ſibi peculiarem, punctatim facili negotio deſignabilem, ità <lb/>conſtrui poſſit, ut unà ſuam naturam ac indolem prodat. </s>
  <s xml:id="echoid-s5200" xml:space="preserve">modus ille <lb/>ſic habet.</s>
  <s xml:id="echoid-s5201" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5202" xml:space="preserve">V. </s>
  <s xml:id="echoid-s5203" xml:space="preserve">Connectatur recta CX; </s>
  <s xml:id="echoid-s5204" xml:space="preserve">fiátque CX. </s>
  <s xml:id="echoid-s5205" xml:space="preserve">CV:</s>
  <s xml:id="echoid-s5206" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s5207" xml:space="preserve">I; </s>
  <s xml:id="echoid-s5208" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5209" xml:space="preserve">per <lb/>
<anchor type="note" xlink:label="note-0099-02a" xlink:href="note-0099-02"/>
punctum V indeſinitè protendatur recta FG, datæ CB parallela; <lb/></s>
  <s xml:id="echoid-s5210" xml:space="preserve">tum è refringentis centro C rectæ quotcunque CI exeant, rectam <lb/>FG decuſſantes punctis H; </s>
  <s xml:id="echoid-s5211" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5212" xml:space="preserve">centro X, intervallo rectas VH per-<lb/>petuùm æquantè deſcripti circuli rectis CI occurrant punctis N; </s>
  <s xml:id="echoid-s5213" xml:space="preserve">per <lb/>hujuſmodi puncta quævis linea tranſit, quam innuimus expoſiti _Pro-_ <lb/>_blematis_ Solutioni deſervituram; </s>
  <s xml:id="echoid-s5214" xml:space="preserve">ejus ſcilicet, &amp; </s>
  <s xml:id="echoid-s5215" xml:space="preserve">dati circuli reſſ@@-<lb/>gentis interſectio quæpiam incidentiæ punctum erit, ad quod per X <lb/>ducta recta refringetur in aliquam ipſi BC parallelam; </s>
  <s xml:id="echoid-s5216" xml:space="preserve">ſeu viciſſim <lb/>hæc in illam. </s>
  <s xml:id="echoid-s5217" xml:space="preserve">Sit enim talis interſectio quævis N; </s>
  <s xml:id="echoid-s5218" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5219" xml:space="preserve">ducta NX ip-<lb/>ſam BC ſecet in K; </s>
  <s xml:id="echoid-s5220" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5221" xml:space="preserve">ſint NM, ac XT ad BC parallelæ. </s>
  <s xml:id="echoid-s5222" xml:space="preserve">Eſtque <lb/>tum CK. </s>
  <s xml:id="echoid-s5223" xml:space="preserve">KN:</s>
  <s xml:id="echoid-s5224" xml:space="preserve">: (TX. </s>
  <s xml:id="echoid-s5225" xml:space="preserve">XN :</s>
  <s xml:id="echoid-s5226" xml:space="preserve">: TX. </s>
  <s xml:id="echoid-s5227" xml:space="preserve">VH :</s>
  <s xml:id="echoid-s5228" xml:space="preserve">: CX. </s>
  <s xml:id="echoid-s5229" xml:space="preserve">CV :</s>
  <s xml:id="echoid-s5230" xml:space="preserve">:) R. </s>
  <s xml:id="echoid-s5231" xml:space="preserve">I; </s>
  <s xml:id="echoid-s5232" xml:space="preserve"><lb/>unde ſecundum oſtenſa liquet NXKrefractum eſſe ipſius MN; </s>
  <s xml:id="echoid-s5233" xml:space="preserve"><lb/>quod oportebat factum. </s>
  <s xml:id="echoid-s5234" xml:space="preserve">Ità _Problema_ δυ{στι}ό{ει}ςον utcunque licebit <lb/>exequi, nec non ejuſce qualitatem intueri; </s>
  <s xml:id="echoid-s5235" xml:space="preserve">quot refracti per ocu-<lb/>li centrum meent definire, ſingulóſque reipsâ deſignare; </s>
  <s xml:id="echoid-s5236" xml:space="preserve">quæ lon-<lb/>giuſculum eſſet ſigillatim exponere. </s>
  <s xml:id="echoid-s5237" xml:space="preserve">cùm autem eâtenus imaginis loeus
<pb o="82" file="0100" n="100" rhead=""/>
habeatur determinatus; </s>
  <s xml:id="echoid-s5238" xml:space="preserve">ſuccedit ut breviter etiam ipſiſſimum in ſin-<lb/>gulo tali refracto punctum oſtendamus, ad quod illa conſiſtit. </s>
  <s xml:id="echoid-s5239" xml:space="preserve">in cu-<lb/>jus rei gratiam hoc quaſi _Lemma_ præſternemus.</s>
  <s xml:id="echoid-s5240" xml:space="preserve"/>
</p>
<div xml:id="echoid-div133" type="float" level="2" n="21">
<note position="right" xlink:label="note-0099-02" xlink:href="note-0099-02a" xml:space="preserve">Fig. 121.</note>
</div>
<p>
  <s xml:id="echoid-s5241" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s5242" xml:space="preserve">In circulo AN B, cujus centrum C, ſint Semidiametro CA <lb/>perpendiculares NE, RF; </s>
  <s xml:id="echoid-s5243" xml:space="preserve">item Semidiametro CB ſint perpendicu-<lb/>lares NG, XH; </s>
  <s xml:id="echoid-s5244" xml:space="preserve">ſint autem CE, EF ipſis CG, GH proportiona-<lb/>
<anchor type="note" xlink:label="note-0100-01a" xlink:href="note-0100-01"/>
les; </s>
  <s xml:id="echoid-s5245" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5246" xml:space="preserve">arcus NR, NX indefinitè parvi; </s>
  <s xml:id="echoid-s5247" xml:space="preserve">ſeu quaſi minimi dictâ <lb/>conditione præditi; </s>
  <s xml:id="echoid-s5248" xml:space="preserve">dicimus arcum NR ad arcum NX rationem ha-<lb/>bere conflatam è rationibus ipſarum CE ad CG, &amp; </s>
  <s xml:id="echoid-s5249" xml:space="preserve">NG ad NE; </s>
  <s xml:id="echoid-s5250" xml:space="preserve">vel eſſe <lb/>arc. </s>
  <s xml:id="echoid-s5251" xml:space="preserve">NR, NX:</s>
  <s xml:id="echoid-s5252" xml:space="preserve">: CE x NG. </s>
  <s xml:id="echoid-s5253" xml:space="preserve">CG x NE. </s>
  <s xml:id="echoid-s5254" xml:space="preserve">Nam per N ducatur VT <lb/>tangens circulum, ipſiſque FR, HX occurrens punctis T, V. </s>
  <s xml:id="echoid-s5255" xml:space="preserve">eſt <lb/>itaque (propter Summam ex Hypotheſi parvitatem dictorum arcuum) <lb/>arc NR. </s>
  <s xml:id="echoid-s5256" xml:space="preserve">CN:</s>
  <s xml:id="echoid-s5257" xml:space="preserve">: NT. </s>
  <s xml:id="echoid-s5258" xml:space="preserve">CN:</s>
  <s xml:id="echoid-s5259" xml:space="preserve">: EF. </s>
  <s xml:id="echoid-s5260" xml:space="preserve">EN. </s>
  <s xml:id="echoid-s5261" xml:space="preserve">item CN. </s>
  <s xml:id="echoid-s5262" xml:space="preserve">arc NX:</s>
  <s xml:id="echoid-s5263" xml:space="preserve">: <lb/>CN. </s>
  <s xml:id="echoid-s5264" xml:space="preserve">NV:</s>
  <s xml:id="echoid-s5265" xml:space="preserve">: NG. </s>
  <s xml:id="echoid-s5266" xml:space="preserve">GH. </s>
  <s xml:id="echoid-s5267" xml:space="preserve">quapropter erit arc NR. </s>
  <s xml:id="echoid-s5268" xml:space="preserve">CN+ CN. <lb/></s>
  <s xml:id="echoid-s5269" xml:space="preserve">arc NX = (EF. </s>
  <s xml:id="echoid-s5270" xml:space="preserve">EN + NG. </s>
  <s xml:id="echoid-s5271" xml:space="preserve">GH = EF. </s>
  <s xml:id="echoid-s5272" xml:space="preserve">GH+ NG. </s>
  <s xml:id="echoid-s5273" xml:space="preserve">EN <lb/> = ) CE. </s>
  <s xml:id="echoid-s5274" xml:space="preserve">CG + NG. </s>
  <s xml:id="echoid-s5275" xml:space="preserve">EN. </s>
  <s xml:id="echoid-s5276" xml:space="preserve">hoc eſt arc NR. </s>
  <s xml:id="echoid-s5277" xml:space="preserve">arc NX = CE. </s>
  <s xml:id="echoid-s5278" xml:space="preserve"><lb/>CG + NG. </s>
  <s xml:id="echoid-s5279" xml:space="preserve">EN: </s>
  <s xml:id="echoid-s5280" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s5281" xml:space="preserve">E. </s>
  <s xml:id="echoid-s5282" xml:space="preserve">D. </s>
  <s xml:id="echoid-s5283" xml:space="preserve">(vel arc NR. </s>
  <s xml:id="echoid-s5284" xml:space="preserve">NX = CE x NG. </s>
  <s xml:id="echoid-s5285" xml:space="preserve"><lb/>CG x EN.)</s>
  <s xml:id="echoid-s5286" xml:space="preserve"/>
</p>
<div xml:id="echoid-div134" type="float" level="2" n="22">
<note position="left" xlink:label="note-0100-01" xlink:href="note-0100-01a" xml:space="preserve">Fig. 122.</note>
</div>
<p>
  <s xml:id="echoid-s5287" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s5288" xml:space="preserve">Sit jam radii cujuſvis talis MNP, refringentem interſecantis <lb/>punctis N, P, refractus N π (refringentem nempe denuò ſecans <lb/>in π) huic autem indeſinitè vicinus (&amp; </s>
  <s xml:id="echoid-s5289" xml:space="preserve">quaſi proximus) adjaceat <lb/>radius QR S, cujus itidem refractus R σ (refringenti nempe rurſus <lb/>occurrens in σ), priorem N π decuſſans in Z; </s>
  <s xml:id="echoid-s5290" xml:space="preserve">biſecentur autem <lb/>ſubtenſæ NP, N π punctis G, E: </s>
  <s xml:id="echoid-s5291" xml:space="preserve">Dico rationem NZ ad GZ com-<lb/>poni è rationibus NG ad NE, &amp; </s>
  <s xml:id="echoid-s5292" xml:space="preserve">CE ad CG.</s>
  <s xml:id="echoid-s5293" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Fig. 123, <lb/>124.</note>
<p>
  <s xml:id="echoid-s5294" xml:space="preserve">Nam ducantur rectæ CE (hæc ipſam RS quoque ſecans in F) &amp; </s>
  <s xml:id="echoid-s5295" xml:space="preserve"><lb/>C G; </s>
  <s xml:id="echoid-s5296" xml:space="preserve">nec non CI ad R σ perpendicularis, &amp; </s>
  <s xml:id="echoid-s5297" xml:space="preserve">in protracta CG ſu-<lb/>@@@@@r CH = CI; </s>
  <s xml:id="echoid-s5298" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5299" xml:space="preserve">per H ducatur XY ad N π parallela, ſeu per-<lb/>pendicularis ad CH; </s>
  <s xml:id="echoid-s5300" xml:space="preserve">unde eſt XY = R π; </s>
  <s xml:id="echoid-s5301" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5302" xml:space="preserve">arc NX = Y π &amp; </s>
  <s xml:id="echoid-s5303" xml:space="preserve"><lb/>arc XY = arc R σ adeóque arc NR ±: </s>
  <s xml:id="echoid-s5304" xml:space="preserve">σ π = 2 arc NX. </s>
  <s xml:id="echoid-s5305" xml:space="preserve">Eſt-<lb/>que prætereà CG. </s>
  <s xml:id="echoid-s5306" xml:space="preserve">CE:</s>
  <s xml:id="echoid-s5307" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s5308" xml:space="preserve">I:</s>
  <s xml:id="echoid-s5309" xml:space="preserve">: CI. </s>
  <s xml:id="echoid-s5310" xml:space="preserve">CF:</s>
  <s xml:id="echoid-s5311" xml:space="preserve">: CH. </s>
  <s xml:id="echoid-s5312" xml:space="preserve">CF; </s>
  <s xml:id="echoid-s5313" xml:space="preserve">adeóque <lb/>permutatim CG. </s>
  <s xml:id="echoid-s5314" xml:space="preserve">CH:</s>
  <s xml:id="echoid-s5315" xml:space="preserve">: CE. </s>
  <s xml:id="echoid-s5316" xml:space="preserve">CF. </s>
  <s xml:id="echoid-s5317" xml:space="preserve">ergò (juxta præcedentem) eſt <lb/>arc. </s>
  <s xml:id="echoid-s5318" xml:space="preserve">NR. </s>
  <s xml:id="echoid-s5319" xml:space="preserve">NX = NG. </s>
  <s xml:id="echoid-s5320" xml:space="preserve">NE + CE. </s>
  <s xml:id="echoid-s5321" xml:space="preserve">CG. </s>
  <s xml:id="echoid-s5322" xml:space="preserve">ad hæc ob illam (quæ <lb/>ponitur) arcuum NR, SP, π σ exiquitatem, erit arc NR. </s>
  <s xml:id="echoid-s5323" xml:space="preserve">π σ :</s>
  <s xml:id="echoid-s5324" xml:space="preserve">: <lb/>ſubtenſa NR. </s>
  <s xml:id="echoid-s5325" xml:space="preserve">π σ :</s>
  <s xml:id="echoid-s5326" xml:space="preserve">: NZ. </s>
  <s xml:id="echoid-s5327" xml:space="preserve">Zσ :</s>
  <s xml:id="echoid-s5328" xml:space="preserve">: NZ. </s>
  <s xml:id="echoid-s5329" xml:space="preserve">Zπ. </s>
  <s xml:id="echoid-s5330" xml:space="preserve">ergò (inverſè compo-<lb/>nendo, vel dividendo, tum &amp; </s>
  <s xml:id="echoid-s5331" xml:space="preserve">conſequentes ſubduplando) arc NR. <lb/></s>
  <s xml:id="echoid-s5332" xml:space="preserve">{arc NR±:</s>
  <s xml:id="echoid-s5333" xml:space="preserve">π σ/2}:</s>
  <s xml:id="echoid-s5334" xml:space="preserve">: NZ.</s>
  <s xml:id="echoid-s5335" xml:space="preserve">{NZ±Zπ/2}. </s>
  <s xml:id="echoid-s5336" xml:space="preserve">atqui velut modò dictum)
<pb o="83" file="0101" n="101" rhead=""/>
arc {NR &amp; </s>
  <s xml:id="echoid-s5337" xml:space="preserve">+ -;</s>
  <s xml:id="echoid-s5338" xml:space="preserve">: πσ/2} = NX; </s>
  <s xml:id="echoid-s5339" xml:space="preserve">item eſt {NZ &amp; </s>
  <s xml:id="echoid-s5340" xml:space="preserve">+ -;</s>
  <s xml:id="echoid-s5341" xml:space="preserve">: Z π/2} = GZ. </s>
  <s xml:id="echoid-s5342" xml:space="preserve">eri@ ergòarc <lb/>
<anchor type="note" xlink:label="note-0101-01a" xlink:href="note-0101-01"/>
NR. </s>
  <s xml:id="echoid-s5343" xml:space="preserve">NX :</s>
  <s xml:id="echoid-s5344" xml:space="preserve">: NZ. </s>
  <s xml:id="echoid-s5345" xml:space="preserve">GZ. </s>
  <s xml:id="echoid-s5346" xml:space="preserve">quapropter erit (juxta præcedentem) <lb/>NZ. </s>
  <s xml:id="echoid-s5347" xml:space="preserve">GZ = NG. </s>
  <s xml:id="echoid-s5348" xml:space="preserve">NE + CE. </s>
  <s xml:id="echoid-s5349" xml:space="preserve">CG.</s>
  <s xml:id="echoid-s5350" xml:space="preserve"/>
</p>
<div xml:id="echoid-div135" type="float" level="2" n="23">
<note position="right" xlink:label="note-0101-01" xlink:href="note-0101-01a" xml:space="preserve">Fig. 124.</note>
</div>
<p>
  <s xml:id="echoid-s5351" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s5352" xml:space="preserve">Porrò liquet punctum Z eſſe locum imaginis, quem expe-<lb/>timus, oculo conſpicuæ in recta N π conſtituto; </s>
  <s xml:id="echoid-s5353" xml:space="preserve">utpote circa quod <lb/>viciniorum ipſi NP radiorum refracti ipſam N π interſecant; </s>
  <s xml:id="echoid-s5354" xml:space="preserve">qua de <lb/>re multoties egimus, ut pigeat eò plura βαττλγ{εĩ}ν.</s>
  <s xml:id="echoid-s5355" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5356" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s5357" xml:space="preserve">Facilè verò, Secundum _Theorema pramiſſum_, deſignatur <lb/>punctum Z. </s>
  <s xml:id="echoid-s5358" xml:space="preserve">Ducatur nempe CG ad refractum NK perpendicula-<lb/>ris; </s>
  <s xml:id="echoid-s5359" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5360" xml:space="preserve">ad connexam CN ducatur perpendicularis GV; </s>
  <s xml:id="echoid-s5361" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5362" xml:space="preserve">per V <lb/>ducatur VZ ad CK parallela, ſecans ipſam NK in Z. </s>
  <s xml:id="echoid-s5363" xml:space="preserve">factum erit. <lb/></s>
  <s xml:id="echoid-s5364" xml:space="preserve">Nam, connexâ GE, liquet angulos GEC, GNC(circumducti <lb/>
<anchor type="note" xlink:label="note-0101-02a" xlink:href="note-0101-02"/>
nempe per N, E, G, C circuli ſubtenfæ GE inſiſtentes ambos) æqua-<lb/>ri; </s>
  <s xml:id="echoid-s5365" xml:space="preserve">hoc eſt angulos GEC, VGCæquari. </s>
  <s xml:id="echoid-s5366" xml:space="preserve">quapropter (utrique <lb/>rectum adjiciendo) toti NEG, ZGVæquantur. </s>
  <s xml:id="echoid-s5367" xml:space="preserve">item alterni <lb/>GNE, VZGæquantur. </s>
  <s xml:id="echoid-s5368" xml:space="preserve">ergò triangula GN E, VZGſimilia ſunt, <lb/>unde NG. </s>
  <s xml:id="echoid-s5369" xml:space="preserve">NE :</s>
  <s xml:id="echoid-s5370" xml:space="preserve">: ZV. </s>
  <s xml:id="echoid-s5371" xml:space="preserve">ZG. </s>
  <s xml:id="echoid-s5372" xml:space="preserve">itaque. </s>
  <s xml:id="echoid-s5373" xml:space="preserve">CE. </s>
  <s xml:id="echoid-s5374" xml:space="preserve">CG + NG. </s>
  <s xml:id="echoid-s5375" xml:space="preserve">NE = <lb/>CE. </s>
  <s xml:id="echoid-s5376" xml:space="preserve">CG + ZV. </s>
  <s xml:id="echoid-s5377" xml:space="preserve">ZG. </s>
  <s xml:id="echoid-s5378" xml:space="preserve">verùm (ob refractionem) eſt NK. </s>
  <s xml:id="echoid-s5379" xml:space="preserve">KC <lb/> :</s>
  <s xml:id="echoid-s5380" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s5381" xml:space="preserve">R :</s>
  <s xml:id="echoid-s5382" xml:space="preserve">: CE. </s>
  <s xml:id="echoid-s5383" xml:space="preserve">CG; </s>
  <s xml:id="echoid-s5384" xml:space="preserve">hoc eſt NZ. </s>
  <s xml:id="echoid-s5385" xml:space="preserve">ZV :</s>
  <s xml:id="echoid-s5386" xml:space="preserve">: CE. </s>
  <s xml:id="echoid-s5387" xml:space="preserve">CG. </s>
  <s xml:id="echoid-s5388" xml:space="preserve">eſt igitur <lb/>CE. </s>
  <s xml:id="echoid-s5389" xml:space="preserve">CG + NG. </s>
  <s xml:id="echoid-s5390" xml:space="preserve">NE = NZ. </s>
  <s xml:id="echoid-s5391" xml:space="preserve">ZV + ZV. </s>
  <s xml:id="echoid-s5392" xml:space="preserve">ZG; </s>
  <s xml:id="echoid-s5393" xml:space="preserve">hoc eſt CE. <lb/></s>
  <s xml:id="echoid-s5394" xml:space="preserve">CG + NG. </s>
  <s xml:id="echoid-s5395" xml:space="preserve">NE = NZ. </s>
  <s xml:id="echoid-s5396" xml:space="preserve">ZG. </s>
  <s xml:id="echoid-s5397" xml:space="preserve">ergò punctum Z conditionem <lb/>obtinet, imaginis loco congruentem, è mox oſtenſis. </s>
  <s xml:id="echoid-s5398" xml:space="preserve">adeò liquet <lb/>propoſitum.</s>
  <s xml:id="echoid-s5399" xml:space="preserve"/>
</p>
<div xml:id="echoid-div136" type="float" level="2" n="24">
<note position="right" xlink:label="note-0101-02" xlink:href="note-0101-02a" xml:space="preserve">Fig. 125.</note>
</div>
<p>
  <s xml:id="echoid-s5400" xml:space="preserve">X. </s>
  <s xml:id="echoid-s5401" xml:space="preserve">Quin ſubnotamus rectam NK ad punctum Z ità dividi, ut ſit <lb/>NZ. </s>
  <s xml:id="echoid-s5402" xml:space="preserve">ZK :</s>
  <s xml:id="echoid-s5403" xml:space="preserve">: NGq. </s>
  <s xml:id="echoid-s5404" xml:space="preserve">CGq. </s>
  <s xml:id="echoid-s5405" xml:space="preserve">Etenim eſt NZ. </s>
  <s xml:id="echoid-s5406" xml:space="preserve">ZK :</s>
  <s xml:id="echoid-s5407" xml:space="preserve">: NV. </s>
  <s xml:id="echoid-s5408" xml:space="preserve">VC <lb/> :</s>
  <s xml:id="echoid-s5409" xml:space="preserve">: NVq. </s>
  <s xml:id="echoid-s5410" xml:space="preserve">VGq :</s>
  <s xml:id="echoid-s5411" xml:space="preserve">: NGq. </s>
  <s xml:id="echoid-s5412" xml:space="preserve">CGq.</s>
  <s xml:id="echoid-s5413" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5414" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s5415" xml:space="preserve">Subjiciam &amp; </s>
  <s xml:id="echoid-s5416" xml:space="preserve">hoc è dictis conſectarium _Theorema:_</s>
  <s xml:id="echoid-s5417" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5418" xml:space="preserve">Fiat √ 3 Rq. </s>
  <s xml:id="echoid-s5419" xml:space="preserve">√ Iq - Rq :</s>
  <s xml:id="echoid-s5420" xml:space="preserve">: CB. </s>
  <s xml:id="echoid-s5421" xml:space="preserve">CQ; </s>
  <s xml:id="echoid-s5422" xml:space="preserve">ductáque QN ad <lb/>CB perpendicularis circumferentiæ occurrat ad N; </s>
  <s xml:id="echoid-s5423" xml:space="preserve">radii verò MN <lb/>ad CB paralleli refractus ſit NK, circuli peripheriæ denuò occur-<lb/>
<anchor type="note" xlink:label="note-0101-03a" xlink:href="note-0101-03"/>
rens in Z; </s>
  <s xml:id="echoid-s5424" xml:space="preserve">dico punctum Z eſſe imaginem, qualem mox definivimus, <lb/>oculo conſpicuam in ipſa NK ſito.</s>
  <s xml:id="echoid-s5425" xml:space="preserve"/>
</p>
<div xml:id="echoid-div137" type="float" level="2" n="25">
<note position="right" xlink:label="note-0101-03" xlink:href="note-0101-03a" xml:space="preserve">Fig. 126.</note>
</div>
<p>
  <s xml:id="echoid-s5426" xml:space="preserve">Nam (ductis CE ad MN, &amp; </s>
  <s xml:id="echoid-s5427" xml:space="preserve">CG ad NZ perpendicularibus, <lb/>ac junctâ CN) ob 3 Rq. </s>
  <s xml:id="echoid-s5428" xml:space="preserve">Iq - Rq :</s>
  <s xml:id="echoid-s5429" xml:space="preserve">: CNq. </s>
  <s xml:id="echoid-s5430" xml:space="preserve">NEq. </s>
  <s xml:id="echoid-s5431" xml:space="preserve">hoc eſt <lb/>3 CGq. </s>
  <s xml:id="echoid-s5432" xml:space="preserve">CEq - CGq :</s>
  <s xml:id="echoid-s5433" xml:space="preserve">: CN q. </s>
  <s xml:id="echoid-s5434" xml:space="preserve">NE q; </s>
  <s xml:id="echoid-s5435" xml:space="preserve">erit dividendo 4 CG q
<pb o="84" file="0102" n="102" rhead=""/>
-CEq. </s>
  <s xml:id="echoid-s5436" xml:space="preserve">CEq - CGq :</s>
  <s xml:id="echoid-s5437" xml:space="preserve">: CNq - NEq. </s>
  <s xml:id="echoid-s5438" xml:space="preserve">NEq :</s>
  <s xml:id="echoid-s5439" xml:space="preserve">: CEq. </s>
  <s xml:id="echoid-s5440" xml:space="preserve">NEq. <lb/></s>
  <s xml:id="echoid-s5441" xml:space="preserve">quare permutando 4 CGq - CEq. </s>
  <s xml:id="echoid-s5442" xml:space="preserve">CEq :</s>
  <s xml:id="echoid-s5443" xml:space="preserve">: CEq - CGq. </s>
  <s xml:id="echoid-s5444" xml:space="preserve"><lb/>NE q. </s>
  <s xml:id="echoid-s5445" xml:space="preserve">(hoc eſt) :</s>
  <s xml:id="echoid-s5446" xml:space="preserve">: NGq - NEq. </s>
  <s xml:id="echoid-s5447" xml:space="preserve">NEq. </s>
  <s xml:id="echoid-s5448" xml:space="preserve">ergò componendo <lb/>4 CGq. </s>
  <s xml:id="echoid-s5449" xml:space="preserve">CEq :</s>
  <s xml:id="echoid-s5450" xml:space="preserve">: NGq. </s>
  <s xml:id="echoid-s5451" xml:space="preserve">NEq. </s>
  <s xml:id="echoid-s5452" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5453" xml:space="preserve">ideò 2 CG. </s>
  <s xml:id="echoid-s5454" xml:space="preserve">CE :</s>
  <s xml:id="echoid-s5455" xml:space="preserve">: NG. </s>
  <s xml:id="echoid-s5456" xml:space="preserve"><lb/>NE. </s>
  <s xml:id="echoid-s5457" xml:space="preserve">quare 2. </s>
  <s xml:id="echoid-s5458" xml:space="preserve">1 + CG. </s>
  <s xml:id="echoid-s5459" xml:space="preserve">CE = NG. </s>
  <s xml:id="echoid-s5460" xml:space="preserve">NE. </s>
  <s xml:id="echoid-s5461" xml:space="preserve">vel 2. </s>
  <s xml:id="echoid-s5462" xml:space="preserve">1 = NG. </s>
  <s xml:id="echoid-s5463" xml:space="preserve"><lb/>NE + CE. </s>
  <s xml:id="echoid-s5464" xml:space="preserve">CG. </s>
  <s xml:id="echoid-s5465" xml:space="preserve">hoc eſt NZ. </s>
  <s xml:id="echoid-s5466" xml:space="preserve">GZ = NG. </s>
  <s xml:id="echoid-s5467" xml:space="preserve">NE + CE. </s>
  <s xml:id="echoid-s5468" xml:space="preserve">CG. </s>
  <s xml:id="echoid-s5469" xml:space="preserve"><lb/>unde liquet, è mox antedictis, propoſitum.</s>
  <s xml:id="echoid-s5470" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5471" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s5472" xml:space="preserve">Ex iſta porrò conſtructione facilè colligitur, ſi fuerit 3 Rq <lb/> = Iq - Rq (hoc eſt ſi 2 R = I) adeóque CQ = CB; </s>
  <s xml:id="echoid-s5473" xml:space="preserve">quòd hu-<lb/>juſmodi punctum Z non aliud erit ab ipſo D; </s>
  <s xml:id="echoid-s5474" xml:space="preserve">ſeu perpendiculari ipſi <lb/>AB debitam imaginem ad punctum D conſiſtere; </s>
  <s xml:id="echoid-s5475" xml:space="preserve">eas verò quæ reli-<lb/>quis refractis conveniunt ejuſmodi imagines intra circulum omnes, vel <lb/>ſupra peripheriam extare. </s>
  <s xml:id="echoid-s5476" xml:space="preserve">quinetiam ſi fuerit 2 R &amp;</s>
  <s xml:id="echoid-s5477" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5478" xml:space="preserve">I, adeoque <lb/>CB &amp;</s>
  <s xml:id="echoid-s5479" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5480" xml:space="preserve">CQ, patet nullius refracti imaginem in peripheria exiſtere, <lb/>ſed omnes ſupra ipſam. </s>
  <s xml:id="echoid-s5481" xml:space="preserve">Enim verò in his caſibus omnes refracti ax-<lb/>em AD ſupra punctum D interſecant. </s>
  <s xml:id="echoid-s5482" xml:space="preserve">verùm ſi fuerit 2 R &amp;</s>
  <s xml:id="echoid-s5483" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5484" xml:space="preserve">I (uti-<lb/>que ſicut reverà quoad pleraſque cunctas in hac rerum natura pelluci-<lb/>das refringentes materias uſu venit) utì reipsâ datur ejuſmodi punctum <lb/>Z, in perepheria TD alicubi ſitum, ità facilè poterit iſto modo de-<lb/>terminari.</s>
  <s xml:id="echoid-s5485" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5486" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s5487" xml:space="preserve">Obſervetur porrò ſic definitum punctum Z circuli partem à <lb/>D verſus T per radios quadranti BT incidentes illuſtratam terminare. <lb/></s>
  <s xml:id="echoid-s5488" xml:space="preserve">Omnes enim ipſo MN obliquiùs incidentium refracti ipſam NZ ſupra <lb/>Z verſus G decuſſabunt; </s>
  <s xml:id="echoid-s5489" xml:space="preserve">adeóque ad partes ZD circulo impingent; </s>
  <s xml:id="echoid-s5490" xml:space="preserve"><lb/>item omnium ipſo MN rectiorum refracti ipſam NZ infra Z verſus K <lb/>interſecabunt; </s>
  <s xml:id="echoid-s5491" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5492" xml:space="preserve">hinc etiam in arcum ZD cadent.</s>
  <s xml:id="echoid-s5493" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5494" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s5495" xml:space="preserve">Exhinc apparet (id quod _ab eximio D. </s>
  <s xml:id="echoid-s5496" xml:space="preserve">Sluſio_ monitum ami-<lb/>
<anchor type="note" xlink:label="note-0102-01a" xlink:href="note-0102-01"/>
cus mihi communicavit) potuiſſe _Carteſium_ ſine tabularum confecti-<lb/>one ſuum _Iridis_ angulum determinare. </s>
  <s xml:id="echoid-s5497" xml:space="preserve">nam aſſumpto arcu DY = DZ; <lb/></s>
  <s xml:id="echoid-s5498" xml:space="preserve">angulum iſtum arcus ZY metitur; </s>
  <s xml:id="echoid-s5499" xml:space="preserve">poſito circulum propoſitum per <lb/>aquei globi centrum tranſire. </s>
  <s xml:id="echoid-s5500" xml:space="preserve">quod ità facilè conſtat. </s>
  <s xml:id="echoid-s5501" xml:space="preserve">Radii cujuſvis <lb/>diametro BC paralleli MN refractus NZKreflectatur in ZF H; </s>
  <s xml:id="echoid-s5502" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s5503" xml:space="preserve">ZF in FO refringatur; </s>
  <s xml:id="echoid-s5504" xml:space="preserve">ſitque FL ad BD parallela; </s>
  <s xml:id="echoid-s5505" xml:space="preserve">ſumatur eti-<lb/>am DY = DZ; </s>
  <s xml:id="echoid-s5506" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5507" xml:space="preserve">connectantur CZ, CY; </s>
  <s xml:id="echoid-s5508" xml:space="preserve">dico angulum LFO <lb/>æquari angulo ZC Y. </s>
  <s xml:id="echoid-s5509" xml:space="preserve">Nam imprimìs ob ZN, ZF æqualiter ad pe-<lb/>ripheriam inclinatos, patet angulum OFHangulo PNZvel CKZ <lb/>æquari. </s>
  <s xml:id="echoid-s5510" xml:space="preserve">igitur ang. </s>
  <s xml:id="echoid-s5511" xml:space="preserve">HFL- HFO = ang FIC- CKZ = ang
<pb o="85" file="0103" n="103" rhead=""/>
KIZ- CKZ = ang NZI- 2 ang CKZ = 2 ang NZ C-<lb/>2 ang CKZ = 2 ang. </s>
  <s xml:id="echoid-s5512" xml:space="preserve">ZCD = ZC Y. </s>
  <s xml:id="echoid-s5513" xml:space="preserve">eſt igitur ang. </s>
  <s xml:id="echoid-s5514" xml:space="preserve">OFL = <lb/>ang ZC Y. </s>
  <s xml:id="echoid-s5515" xml:space="preserve">Cùm itaque ſit in ſuperiore Hypotheſi punctum Z um-<lb/>bræ lucíſque confinium, manifeſtè liquet propoſitum.</s>
  <s xml:id="echoid-s5516" xml:space="preserve"/>
</p>
<div xml:id="echoid-div138" type="float" level="2" n="26">
<note position="left" xlink:label="note-0102-01" xlink:href="note-0102-01a" xml:space="preserve">Fig. 127.</note>
</div>
<p>
  <s xml:id="echoid-s5517" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s5518" xml:space="preserve">Subnotetur autem, ſi medium inflectens ſit aqueum, arcum <lb/>ZY eſſe partem circuli totam (poſticam ſcilicet) illuminatam; </s>
  <s xml:id="echoid-s5519" xml:space="preserve">tan-<lb/>gentis enim ST refractus, puta TV, nedum non punctum Y præ-<lb/>tergreditur, at citra punctum D cadit. </s>
  <s xml:id="echoid-s5520" xml:space="preserve">aſt in denſioribus mediis, ve-<lb/>lut in vitro, ſecùs accidere poteſt; </s>
  <s xml:id="echoid-s5521" xml:space="preserve">ſiquidem in eo tangentis refractus, <lb/>
<anchor type="note" xlink:label="note-0103-01a" xlink:href="note-0103-01"/>
puta TX, ultra terminum Y (modo prædicto deſignatum) cadit, <lb/>ut quidem ex calculo facilè colligatur; </s>
  <s xml:id="echoid-s5522" xml:space="preserve">unde pars illuminata arcu ZY <lb/>amplior evadit; </s>
  <s xml:id="echoid-s5523" xml:space="preserve">tangentium quippe refractis circumſcripta. </s>
  <s xml:id="echoid-s5524" xml:space="preserve">Vide-<lb/>rit igitur excellentiſſimus vir; </s>
  <s xml:id="echoid-s5525" xml:space="preserve">an univerſim conftet (id quod ipſe <lb/>niſi fallor innuere videbatur) ex obſervata partis illuminatæ quantitate, <lb/>_Iridis angulam, etiam juxta Carteſianas Hypotbeſes, recte determi-_ <lb/>_nari._ </s>
  <s xml:id="echoid-s5526" xml:space="preserve">Nam ſumendo arcum DR = DX; </s>
  <s xml:id="echoid-s5527" xml:space="preserve">ad punctum quidem R <lb/>pertinget illuſtratio; </s>
  <s xml:id="echoid-s5528" xml:space="preserve">neque tamen ulla lux quadranti BT incidens à <lb/>parte ZR (ſed illa tantùm quæ ad partes ZD cadit) ad oculum O <lb/>inflectetur. </s>
  <s xml:id="echoid-s5529" xml:space="preserve">unde quoad oculos ad has partes ſitos, hoc eſt quoad rem <lb/>quæ præ manibus, punctum Z lucem &amp; </s>
  <s xml:id="echoid-s5530" xml:space="preserve">umbram dirimit atque diſter-<lb/>minat.</s>
  <s xml:id="echoid-s5531" xml:space="preserve"/>
</p>
<div xml:id="echoid-div139" type="float" level="2" n="27">
<note position="right" xlink:label="note-0103-01" xlink:href="note-0103-01a" xml:space="preserve">Fig. 128.</note>
</div>
<p>
  <s xml:id="echoid-s5532" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s5533" xml:space="preserve">Vobis autem expendendum propono, annon exhinc _appa-_ <lb/>
<anchor type="note" xlink:label="note-0103-02a" xlink:href="note-0103-02"/>
_rentiarum in Iride ratio_ elici poſſit, illâ fortè veriſimi ior, quam <lb/>ipſe _Carteſius_ aſſignavit. </s>
  <s xml:id="echoid-s5534" xml:space="preserve">quid enim ſi dixero peripheriæ ZV impin-<lb/>gentem lucem, &amp; </s>
  <s xml:id="echoid-s5535" xml:space="preserve">verſus O inflexam magìs apparere; </s>
  <s xml:id="echoid-s5536" xml:space="preserve">primò, quia ſpiſ-<lb/>ſior eſt, ac à radiorum geminâ diffuſione conſtat, ab utraque puncti N <lb/>parte in arcum ZV retractorum; </s>
  <s xml:id="echoid-s5537" xml:space="preserve">tum ſecundò quoniam obliquius <lb/>ipſi ZV incidit, adeóque faciliùs &amp; </s>
  <s xml:id="echoid-s5538" xml:space="preserve">copioſiùs indè quam aliunde <lb/>verſus partes O retorquetur? </s>
  <s xml:id="echoid-s5539" xml:space="preserve">Et cum præſertim circa punctum Z <lb/>acutiùs radii coëant, neque non incurrant obliquius; </s>
  <s xml:id="echoid-s5540" xml:space="preserve">quidni proptereà <lb/>vividier exindè reſultet apparentia? </s>
  <s xml:id="echoid-s5541" xml:space="preserve">Verùm hæc παρβακῶ<unsure/>ς.</s>
  <s xml:id="echoid-s5542" xml:space="preserve"/>
</p>
<div xml:id="echoid-div140" type="float" level="2" n="28">
<note position="right" xlink:label="note-0103-02" xlink:href="note-0103-02a" xml:space="preserve">Fig. 128.</note>
</div>
<p>
  <s xml:id="echoid-s5543" xml:space="preserve">_Quoniam Colorum incidit mentio, quid ſi de illis_ (etſi præter morem <lb/>ac ordinem) paucula divinavero?</s>
  <s xml:id="echoid-s5544" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5545" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s5546" xml:space="preserve">_Album_ eſt quod lucem copioſam, pariter ubique ſpiſſam, <lb/>circumfundit. </s>
  <s xml:id="echoid-s5547" xml:space="preserve">Talia fermè ſunt corpora, rarioribus poris inter-<lb/>puncta; </s>
  <s xml:id="echoid-s5548" xml:space="preserve">præſertim, quæ multas ſuperficieculas, in omne latus obver-<lb/>ſas, habent. </s>
  <s xml:id="echoid-s5549" xml:space="preserve">Suadetur hoc, Quia purè lucida ſemper alba videntur;</s>
  <s xml:id="echoid-s5550" xml:space="preserve">
<pb o="86" file="0104" n="104" rhead=""/>
Quia corpus bene terſum luci Splendidæ expoſitum albeſcit; </s>
  <s xml:id="echoid-s5551" xml:space="preserve">Quo-<lb/>niam alba difficiliùs ignem concipiunt; </s>
  <s xml:id="echoid-s5552" xml:space="preserve">Quòd humore tenuiore vacua-<lb/>ta corpora (_Capilli, Polia, Cineres) canitiem_ acquirunt; </s>
  <s xml:id="echoid-s5553" xml:space="preserve">Quibus &amp; </s>
  <s xml:id="echoid-s5554" xml:space="preserve"><lb/>frigore conſtricta accenſeri poſſent.</s>
  <s xml:id="echoid-s5555" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5556" xml:space="preserve">_Nigrum_ eſt, quod lucem minimè, vel parciſimè refundit. </s>
  <s xml:id="echoid-s5557" xml:space="preserve">talia ple-<lb/>runque ſunt corpora valdè pellucida; </s>
  <s xml:id="echoid-s5558" xml:space="preserve">nec non quæ crebros meatus, <lb/>&amp; </s>
  <s xml:id="echoid-s5559" xml:space="preserve">cavernulas lucem abſorbentes habent. </s>
  <s xml:id="echoid-s5560" xml:space="preserve">Hoc indicat, Quod omnes <lb/>_Umbrœ nigrœ apparent; </s>
  <s xml:id="echoid-s5561" xml:space="preserve">Quod Aqua, Vitrum, Nubes_ ad hunc colo-<lb/>rem vergunt; </s>
  <s xml:id="echoid-s5562" xml:space="preserve">Quod _nigra_ faciliùs ignem imbibunt, calefiunt, com-<lb/>buruntur; </s>
  <s xml:id="echoid-s5563" xml:space="preserve">Quòd longiùs diſſita (quorum ſenſim intercipitur, &amp; </s>
  <s xml:id="echoid-s5564" xml:space="preserve">amit-<lb/>titur lux) obſcuriora videntur.</s>
  <s xml:id="echoid-s5565" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5566" xml:space="preserve">_Rubrum eſt_, quod lucem eſſundit hinc indè confertam, ac ſolito <lb/>
<anchor type="note" xlink:label="note-0104-01a" xlink:href="note-0104-01"/>
magìs conſtipatam, aſt interſtitiis umbroſis diremptam, &amp; </s>
  <s xml:id="echoid-s5567" xml:space="preserve">interruptam. <lb/></s>
  <s xml:id="echoid-s5568" xml:space="preserve">talia concipi poſſunt corpora, multas intra ſe quaſi _fornaculas &amp; </s>
  <s xml:id="echoid-s5569" xml:space="preserve">focos_ <lb/>habentia (qualia X, è _Speculis cavis_ contextum; </s>
  <s xml:id="echoid-s5570" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5571" xml:space="preserve">Y è _Sphœrulis_, <lb/>tranſmiſſam lucem ad totidem _focas_ cogentibus, conſtans). </s>
  <s xml:id="echoid-s5572" xml:space="preserve">Argu-<lb/>mento ſit, Quòd à _Speculis, &amp; </s>
  <s xml:id="echoid-s5573" xml:space="preserve">vitris uſtoriis_ collecta lux rubeſcit; </s>
  <s xml:id="echoid-s5574" xml:space="preserve"><lb/>Quòd corpora denſa ignita (quippe quorum cellæ luce ſpiſsâ refer-<lb/>ciuntur) rubra videntur; </s>
  <s xml:id="echoid-s5575" xml:space="preserve">Quòd roſcida nubes Soli (matutino, vel <lb/>veſpertino) expoſita rubet; </s>
  <s xml:id="echoid-s5576" xml:space="preserve">Quòd eroſio rubiginem parit.</s>
  <s xml:id="echoid-s5577" xml:space="preserve">‖ Ad <lb/>rubri naturam fortaſſe pertinet, quòd compreſſa lux languidiùs emicat.</s>
  <s xml:id="echoid-s5578" xml:space="preserve"/>
</p>
<div xml:id="echoid-div141" type="float" level="2" n="29">
<note position="left" xlink:label="note-0104-01" xlink:href="note-0104-01a" xml:space="preserve">@ Lat. <lb/>Fig. 128.</note>
</div>
<p>
  <s xml:id="echoid-s5579" xml:space="preserve">_Cœruleum_ eſt quod lucem raram, aut impetu ſegniore concitatam <lb/>emittit. </s>
  <s xml:id="echoid-s5580" xml:space="preserve">talia videntur eſſe corpora, quæ particulis conſtant albis ac <lb/>atris alternatim diſpoſitis; </s>
  <s xml:id="echoid-s5581" xml:space="preserve">ſed &amp; </s>
  <s xml:id="echoid-s5582" xml:space="preserve">hunc ſubinde colorem oſtentant can-<lb/>dida maligniùs illuſtrata. </s>
  <s xml:id="echoid-s5583" xml:space="preserve">Exemplo ſint, _Æther Sudus_ (in quo <lb/>nempe pauciora natant corpuſcula lucem ad oculos reverberantia, cæ-<lb/>terâ luce dilabente) _Mare_, ſale candido nimirum &amp; </s>
  <s xml:id="echoid-s5584" xml:space="preserve">humore pellu-<lb/>cido conſtans; </s>
  <s xml:id="echoid-s5585" xml:space="preserve">Umbra corporis cujuſvis opaci, de die, ad lucernam <lb/>ardentem facta, &amp; </s>
  <s xml:id="echoid-s5586" xml:space="preserve">ad chartam albam excepta ſeu terminata; </s>
  <s xml:id="echoid-s5587" xml:space="preserve">(nempe <lb/>corporis AB ad chartam XY violacea depingitur umbra, à lucerna C).</s>
  <s xml:id="echoid-s5588" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5589" xml:space="preserve">_Viride_ cæruleo perquàm agnatum eſt. </s>
  <s xml:id="echoid-s5590" xml:space="preserve">_Diſcrimen_ explorent ſaga-<lb/>
<anchor type="note" xlink:label="note-0104-02a" xlink:href="note-0104-02"/>
ciores; </s>
  <s xml:id="echoid-s5591" xml:space="preserve">ego non auſim ariolari.</s>
  <s xml:id="echoid-s5592" xml:space="preserve"/>
</p>
<div xml:id="echoid-div142" type="float" level="2" n="30">
<note position="left" xlink:label="note-0104-02" xlink:href="note-0104-02a" xml:space="preserve">@ Lat. <lb/>Fig. 128.</note>
</div>
<p>
  <s xml:id="echoid-s5593" xml:space="preserve">Cæterùm reliqua colorata ex iſtis variè commixtis, atque contem-<lb/>peratis emergunt; </s>
  <s xml:id="echoid-s5594" xml:space="preserve">ut _flavum_ ex albo copioſo, rubríque nonnihillo in-<lb/>terſperſo; </s>
  <s xml:id="echoid-s5595" xml:space="preserve">_purputeum_ ex multo cæruleo, rubríque tantillo, &amp;</s>
  <s xml:id="echoid-s5596" xml:space="preserve">c. </s>
  <s xml:id="echoid-s5597" xml:space="preserve">Ve-<lb/>rùm ſuſficiat hâctenus, iſta ſupra captum noſtrum poſita Scrutantes; <lb/></s>
  <s xml:id="echoid-s5598" xml:space="preserve">nos illis, quiàηιολογſicas pbyſicas moroſiùs excipiunt, deridendos pro-<lb/>pinaſſe.</s>
  <s xml:id="echoid-s5599" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5600" xml:space="preserve">Sufficient hæc pro radiis parallelis; </s>
  <s xml:id="echoid-s5601" xml:space="preserve">ad divergentes ordine proce-<lb/>dendum eſt; </s>
  <s xml:id="echoid-s5602" xml:space="preserve">aſt interpoſitâ morâ, nè vix exorſi cogamur abrumpere.</s>
  <s xml:id="echoid-s5603" xml:space="preserve">‖</s>
</p>
<pb o="87" file="0105" n="105" rhead=""/>
<p>
  <s xml:id="echoid-s5604" xml:space="preserve">1._</s>
  <s xml:id="echoid-s5605" xml:space="preserve">Tranſactis iis quœ refractioni conveniunt iſti, quam ad ctr_-<lb/>_culi peripberiam ſubeunt radii ſibimet paralleli; </s>
  <s xml:id="echoid-s5606" xml:space="preserve">quid iis. </s>
  <s xml:id="echoid-s5607" xml:space="preserve">ob_-<lb/>_venit proximè diſpicien ſxm venit, qui a puncto qu@piam ſenſibiliter_ <lb/>_diver gentes itidem circulo ſe objiciunt refringendos._ </s>
  <s xml:id="echoid-s5608" xml:space="preserve">cum autem in hac <lb/>Hypotheſi multa reperiatur caſuum varietus è pluribus cauſis oriunda <lb/>(nedum enim à mediorum ſpecie differentium ordine, vel ſitu verſus <lb/>ſe diverſo; </s>
  <s xml:id="echoid-s5609" xml:space="preserve">quinetiam circuli refringentis alia ac alia, convexa nem-<lb/>pe vel concava, facie radiationi obverſa; </s>
  <s xml:id="echoid-s5610" xml:space="preserve">ſed ab ipſius quoque radi-<lb/>antis magìs aut minùs à refringente ſemoti poſitione concluſionum <lb/>emergit nonnulla diſcrepantia) nobis incumbet ità rem, quâ poſſu-<lb/>mus, moderari, ſimul ut cùm ex abſtractione nimia proveniens con-<lb/>fuſio, tum è repetitione faſtidium aliquouſque devitentur. </s>
  <s xml:id="echoid-s5611" xml:space="preserve">id autem <lb/>non aliàs, opinor, commodiùs aſſequemur quàm imprimìs generalia <lb/>quædam attingendo, cuidam uni caſui (illi nempe, ubi I &amp;</s>
  <s xml:id="echoid-s5612" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5613" xml:space="preserve">R, &amp; </s>
  <s xml:id="echoid-s5614" xml:space="preserve"><lb/>radii convexis circuli partibus incidunt). </s>
  <s xml:id="echoid-s5615" xml:space="preserve">Sic applicata, ut ſatì; </s>
  <s xml:id="echoid-s5616" xml:space="preserve">faci-<lb/>lè poſſint ad alios quoque transferri; </s>
  <s xml:id="echoid-s5617" xml:space="preserve">tum peculiaria nonnulla ſingulis <lb/>congruentia ſubnotando. </s>
  <s xml:id="echoid-s5618" xml:space="preserve">ad rem.</s>
  <s xml:id="echoid-s5619" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5620" xml:space="preserve">II. </s>
  <s xml:id="echoid-s5621" xml:space="preserve">In circulum refringentem BN (cujus centrum C) radiet pun-<lb/>
<anchor type="note" xlink:label="note-0105-01a" xlink:href="note-0105-01"/>
ctum A; </s>
  <s xml:id="echoid-s5622" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5623" xml:space="preserve">connexa A @ protendatur ad utraſque partcs indefinitè; <lb/></s>
  <s xml:id="echoid-s5624" xml:space="preserve">tum cujuſvis incidentis AN ſit refractus NK, cum axe nimirum in K <lb/>conveniens; </s>
  <s xml:id="echoid-s5625" xml:space="preserve">dico compoſitas rationes AC ad CK, &amp; </s>
  <s xml:id="echoid-s5626" xml:space="preserve">NK ad NA <lb/>æquari rationi I ad R. </s>
  <s xml:id="echoid-s5627" xml:space="preserve">Conjungatur enim CN, &amp; </s>
  <s xml:id="echoid-s5628" xml:space="preserve">ducatur KH ad <lb/>CN parallela; </s>
  <s xml:id="echoid-s5629" xml:space="preserve">erit igitur (ut generatim antehac habetur oſtenſum) <lb/>I. </s>
  <s xml:id="echoid-s5630" xml:space="preserve">R :</s>
  <s xml:id="echoid-s5631" xml:space="preserve">: NK. </s>
  <s xml:id="echoid-s5632" xml:space="preserve">NH = NK. </s>
  <s xml:id="echoid-s5633" xml:space="preserve">NA + NA. </s>
  <s xml:id="echoid-s5634" xml:space="preserve">NH = NK. </s>
  <s xml:id="echoid-s5635" xml:space="preserve">NA + <lb/>
<anchor type="note" xlink:label="note-0105-02a" xlink:href="note-0105-02"/>
AC. </s>
  <s xml:id="echoid-s5636" xml:space="preserve">CK: </s>
  <s xml:id="echoid-s5637" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s5638" xml:space="preserve">E. </s>
  <s xml:id="echoid-s5639" xml:space="preserve">D.</s>
  <s xml:id="echoid-s5640" xml:space="preserve"/>
</p>
<div xml:id="echoid-div143" type="float" level="2" n="31">
<note position="right" xlink:label="note-0105-01" xlink:href="note-0105-01a" xml:space="preserve">Fig. 129.</note>
<note position="right" xlink:label="note-0105-02" xlink:href="note-0105-02a" xml:space="preserve">3 Lect. num. 9@</note>
</div>
<p>
  <s xml:id="echoid-s5641" xml:space="preserve">III. </s>
  <s xml:id="echoid-s5642" xml:space="preserve">Hinc ſi fuerit CA. </s>
  <s xml:id="echoid-s5643" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s5644" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s5645" xml:space="preserve">R. </s>
  <s xml:id="echoid-s5646" xml:space="preserve">erit CK, CR :</s>
  <s xml:id="echoid-s5647" xml:space="preserve">: NK. <lb/></s>
  <s xml:id="echoid-s5648" xml:space="preserve">NA.</s>
  <s xml:id="echoid-s5649" xml:space="preserve"/>
</p>
<pb o="88" file="0106" n="106" rhead=""/>
<p>
  <s xml:id="echoid-s5650" xml:space="preserve">Nam erit tum CA. </s>
  <s xml:id="echoid-s5651" xml:space="preserve">CR = CA. </s>
  <s xml:id="echoid-s5652" xml:space="preserve">CK + NK. </s>
  <s xml:id="echoid-s5653" xml:space="preserve">NA. </s>
  <s xml:id="echoid-s5654" xml:space="preserve">unde, com-<lb/>munem utrinque adjiciendo rationem CK ad CA, erit CK. </s>
  <s xml:id="echoid-s5655" xml:space="preserve">CA <lb/> + CA. </s>
  <s xml:id="echoid-s5656" xml:space="preserve">CR = CA. </s>
  <s xml:id="echoid-s5657" xml:space="preserve">CK + CK. </s>
  <s xml:id="echoid-s5658" xml:space="preserve">CA + NK. </s>
  <s xml:id="echoid-s5659" xml:space="preserve">NA. </s>
  <s xml:id="echoid-s5660" xml:space="preserve">hoc eſt <lb/>CK. </s>
  <s xml:id="echoid-s5661" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s5662" xml:space="preserve">: NK. </s>
  <s xml:id="echoid-s5663" xml:space="preserve">NA.</s>
  <s xml:id="echoid-s5664" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5665" xml:space="preserve">Notetur in figuris ſequentibus eſſe perpetuo CA. </s>
  <s xml:id="echoid-s5666" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s5667" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s5668" xml:space="preserve">R. <lb/></s>
  <s xml:id="echoid-s5669" xml:space="preserve">
<anchor type="note" xlink:label="note-0106-01a" xlink:href="note-0106-01"/>
quod ſemel, brevitatis causâ, monitum efto.</s>
  <s xml:id="echoid-s5670" xml:space="preserve"/>
</p>
<div xml:id="echoid-div144" type="float" level="2" n="32">
<note position="left" xlink:label="note-0106-01" xlink:href="note-0106-01a" xml:space="preserve">Fig. 130, <lb/>131.</note>
</div>
<p>
  <s xml:id="echoid-s5671" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s5672" xml:space="preserve">Hinc conſectatur; </s>
  <s xml:id="echoid-s5673" xml:space="preserve">primò; </s>
  <s xml:id="echoid-s5674" xml:space="preserve">Si fuerit AN &amp;</s>
  <s xml:id="echoid-s5675" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5676" xml:space="preserve">CR, quòd re-<lb/>fractus N _a_ cum axe AC prorsùm excurrens conveniet. </s>
  <s xml:id="echoid-s5677" xml:space="preserve">Nam erit <lb/>
<anchor type="note" xlink:label="note-0106-02a" xlink:href="note-0106-02"/>
CK. </s>
  <s xml:id="echoid-s5678" xml:space="preserve">AN &amp;</s>
  <s xml:id="echoid-s5679" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5680" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s5681" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s5682" xml:space="preserve">: NK. </s>
  <s xml:id="echoid-s5683" xml:space="preserve">AN. </s>
  <s xml:id="echoid-s5684" xml:space="preserve">adeóque CK &amp;</s>
  <s xml:id="echoid-s5685" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5686" xml:space="preserve">NK.</s>
  <s xml:id="echoid-s5687" xml:space="preserve"/>
</p>
<div xml:id="echoid-div145" type="float" level="2" n="33">
<note position="left" xlink:label="note-0106-02" xlink:href="note-0106-02a" xml:space="preserve">Fig. 129.</note>
</div>
<p>
  <s xml:id="echoid-s5688" xml:space="preserve">V. </s>
  <s xml:id="echoid-s5689" xml:space="preserve">Secundò; </s>
  <s xml:id="echoid-s5690" xml:space="preserve">ſi fuerit AN = CR, refractus N _a_ ad AC pa-<lb/>
<anchor type="note" xlink:label="note-0106-03a" xlink:href="note-0106-03"/>
rallelus erit.</s>
  <s xml:id="echoid-s5691" xml:space="preserve"/>
</p>
<div xml:id="echoid-div146" type="float" level="2" n="34">
<note position="left" xlink:label="note-0106-03" xlink:href="note-0106-03a" xml:space="preserve">Fig. 130.</note>
</div>
<p>
  <s xml:id="echoid-s5692" xml:space="preserve">Nam ſit NH ad AC parallela. </s>
  <s xml:id="echoid-s5693" xml:space="preserve">quum itaque ſit CA. </s>
  <s xml:id="echoid-s5694" xml:space="preserve">AN :</s>
  <s xml:id="echoid-s5695" xml:space="preserve">: <lb/>(CA. </s>
  <s xml:id="echoid-s5696" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s5697" xml:space="preserve">: ) I. </s>
  <s xml:id="echoid-s5698" xml:space="preserve">R; </s>
  <s xml:id="echoid-s5699" xml:space="preserve">erit AN ipſius HN refractus. </s>
  <s xml:id="echoid-s5700" xml:space="preserve">ergò viciſſim <lb/>N Hipſius AN.</s>
  <s xml:id="echoid-s5701" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5702" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s5703" xml:space="preserve">Tertiò; </s>
  <s xml:id="echoid-s5704" xml:space="preserve">Si fuerit AN &amp;</s>
  <s xml:id="echoid-s5705" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5706" xml:space="preserve">CR, refractus N _a_ cum AC <lb/>
<anchor type="note" xlink:label="note-0106-04a" xlink:href="note-0106-04"/>
retrò conveniet extractus.</s>
  <s xml:id="echoid-s5707" xml:space="preserve"/>
</p>
<div xml:id="echoid-div147" type="float" level="2" n="35">
<note position="left" xlink:label="note-0106-04" xlink:href="note-0106-04a" xml:space="preserve">Fig. 131.</note>
</div>
<p>
  <s xml:id="echoid-s5708" xml:space="preserve">Erit enim tunc CK. </s>
  <s xml:id="echoid-s5709" xml:space="preserve">AN &amp;</s>
  <s xml:id="echoid-s5710" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5711" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s5712" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s5713" xml:space="preserve">: NK. </s>
  <s xml:id="echoid-s5714" xml:space="preserve">AN. </s>
  <s xml:id="echoid-s5715" xml:space="preserve">ac indè <lb/>CK &amp;</s>
  <s xml:id="echoid-s5716" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5717" xml:space="preserve">NK.</s>
  <s xml:id="echoid-s5718" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5719" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s5720" xml:space="preserve">Hinc clarum eſt; </s>
  <s xml:id="echoid-s5721" xml:space="preserve">Si fuerit AB non minor quàm CR, om-<lb/>nes refractos verſus AC procurrentes convergere. </s>
  <s xml:id="echoid-s5722" xml:space="preserve">erit enim tunc <lb/>ſemper AN &amp;</s>
  <s xml:id="echoid-s5723" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5724" xml:space="preserve">CR.</s>
  <s xml:id="echoid-s5725" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5726" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s5727" xml:space="preserve">Subnotetur autem ſi fuerit ſaltem AB = CR; </s>
  <s xml:id="echoid-s5728" xml:space="preserve">axi propio-<lb/>res radios in ſenſibilem paralleliſmum refringi.</s>
  <s xml:id="echoid-s5729" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5730" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s5731" xml:space="preserve">Item, Si AT circulum tangat, &amp; </s>
  <s xml:id="echoid-s5732" xml:space="preserve">fuerit AT &amp;</s>
  <s xml:id="echoid-s5733" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5734" xml:space="preserve">CR; </s>
  <s xml:id="echoid-s5735" xml:space="preserve">ma-<lb/>nifeſtum eſt omnes refractos retrò protractos cum AC concurrere. <lb/></s>
  <s xml:id="echoid-s5736" xml:space="preserve">tunc enim ſemper eſt AN &amp;</s>
  <s xml:id="echoid-s5737" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5738" xml:space="preserve">CR.</s>
  <s xml:id="echoid-s5739" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5740" xml:space="preserve">X. </s>
  <s xml:id="echoid-s5741" xml:space="preserve">Clarum eſt quoque, ſi AN = CR, omnes arcui BN inci-<lb/>dentium refractos retrò productos, omnes autem arcui NT inciden-<lb/>tium refractos antrorſum procurrentes axi occurrere.</s>
  <s xml:id="echoid-s5742" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5743" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s5744" xml:space="preserve">Quum autem in caſu, propoſiti maximè contrario (quum <lb/>nempe I &amp;</s>
  <s xml:id="echoid-s5745" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5746" xml:space="preserve">R; </s>
  <s xml:id="echoid-s5747" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5748" xml:space="preserve">radii concavis incidunt partibus) adſimilis contingat <lb/>diverſitas, hanc quoqne breviter attingemus.</s>
  <s xml:id="echoid-s5749" xml:space="preserve"/>
</p>
<pb o="89" file="0107" n="107" rhead=""/>
<p>
  <s xml:id="echoid-s5750" xml:space="preserve">1. </s>
  <s xml:id="echoid-s5751" xml:space="preserve">Si fuerit AN &amp;</s>
  <s xml:id="echoid-s5752" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5753" xml:space="preserve">CR, refractus N _a_ cum AC retrò tractus <lb/>
<anchor type="note" xlink:label="note-0107-01a" xlink:href="note-0107-01"/>
conveniet.</s>
  <s xml:id="echoid-s5754" xml:space="preserve"/>
</p>
<div xml:id="echoid-div148" type="float" level="2" n="36">
<note position="right" xlink:label="note-0107-01" xlink:href="note-0107-01a" xml:space="preserve">Fig. 132.</note>
</div>
<p>
  <s xml:id="echoid-s5755" xml:space="preserve">Nam CK&amp;</s>
  <s xml:id="echoid-s5756" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5757" xml:space="preserve">NK. </s>
  <s xml:id="echoid-s5758" xml:space="preserve">(utin priore caſu).</s>
  <s xml:id="echoid-s5759" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5760" xml:space="preserve">2. </s>
  <s xml:id="echoid-s5761" xml:space="preserve">Etiam hîc ſi AN = CR, refractus N _a_ fit ipſi AC paralle-<lb/>lus.</s>
  <s xml:id="echoid-s5762" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5763" xml:space="preserve">Nam erit CK = NK. </s>
  <s xml:id="echoid-s5764" xml:space="preserve">quod in hoc caſu niſi K infinitè diſtet con-<lb/>tingere nequit.</s>
  <s xml:id="echoid-s5765" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5766" xml:space="preserve">3. </s>
  <s xml:id="echoid-s5767" xml:space="preserve">Si AN&amp;</s>
  <s xml:id="echoid-s5768" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5769" xml:space="preserve">CR; </s>
  <s xml:id="echoid-s5770" xml:space="preserve">refractus N _a_ prorsùm excurrens axi oc-<lb/>currit.</s>
  <s xml:id="echoid-s5771" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5772" xml:space="preserve">Nam hîc CK&amp;</s>
  <s xml:id="echoid-s5773" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5774" xml:space="preserve">NK.</s>
  <s xml:id="echoid-s5775" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5776" xml:space="preserve">4. </s>
  <s xml:id="echoid-s5777" xml:space="preserve">Si AB&amp;</s>
  <s xml:id="echoid-s5778" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5779" xml:space="preserve">CR; </s>
  <s xml:id="echoid-s5780" xml:space="preserve">omnes refracti directè progredientes ad AC <lb/>
<anchor type="note" xlink:label="note-0107-02a" xlink:href="note-0107-02"/>
convergunt. </s>
  <s xml:id="echoid-s5781" xml:space="preserve">Erit enim quivis incidens AN&amp;</s>
  <s xml:id="echoid-s5782" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5783" xml:space="preserve">CR.</s>
  <s xml:id="echoid-s5784" xml:space="preserve"/>
</p>
<div xml:id="echoid-div149" type="float" level="2" n="37">
<note position="right" xlink:label="note-0107-02" xlink:href="note-0107-02a" xml:space="preserve">Fig. 133.</note>
</div>
<p>
  <s xml:id="echoid-s5785" xml:space="preserve">5. </s>
  <s xml:id="echoid-s5786" xml:space="preserve">Quum AN = CR, evidens eſt omnes arcui BN incidentes <lb/>retrorſum verſus CA refractos convergere; </s>
  <s xml:id="echoid-s5787" xml:space="preserve">omnes autem ad partes <lb/>NT cadentes antrorſum verſus CB refringi.</s>
  <s xml:id="echoid-s5788" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5789" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s5790" xml:space="preserve">Hinc apparet ſub iſtis duobus generalibus caſibus tres à diverſo <lb/>
<anchor type="note" xlink:label="note-0107-03a" xlink:href="note-0107-03"/>
puncti radiantis intervallo ſubnaſcentes ſpeciales caſus comprehendi; <lb/></s>
  <s xml:id="echoid-s5791" xml:space="preserve">nempe vel omnes ab axe poſt refractionem progredientes divergunt, <lb/>vel omnes ad ipſum convergunt, vel aliqui divergunt, alii convergunt, <lb/>his intercedente medio quodam ad illum parallelo. </s>
  <s xml:id="echoid-s5792" xml:space="preserve">quæ ſubnotâſſe <lb/>diſcrimina videbatur operæ pretium ac determinâſſe. </s>
  <s xml:id="echoid-s5793" xml:space="preserve">Subdimus <lb/>etiam quoad reliquos generales caſus ſimpliciùs ſeſe rem habere; </s>
  <s xml:id="echoid-s5794" xml:space="preserve"><lb/>ſcilicet eodem ſemper modo: </s>
  <s xml:id="echoid-s5795" xml:space="preserve">Omnes enim ad cavum denſius inci-<lb/>dentium refracti directè procedentes ab axe divergunt; </s>
  <s xml:id="echoid-s5796" xml:space="preserve">Ut &amp; </s>
  <s xml:id="echoid-s5797" xml:space="preserve">omnes <lb/>eorum, qui convexo ratiori impungunt; </s>
  <s xml:id="echoid-s5798" xml:space="preserve">id quod è generaliſſimis re-<lb/>fractionum legibus immediatè ſequitur, &amp; </s>
  <s xml:id="echoid-s5799" xml:space="preserve">è ſimplice ſecundum illas <lb/>linearum ductu diluceſcit. </s>
  <s xml:id="echoid-s5800" xml:space="preserve">His admonitis in orbitam regreſſi pergi-<lb/>mus.</s>
  <s xml:id="echoid-s5801" xml:space="preserve"/>
</p>
<div xml:id="echoid-div150" type="float" level="2" n="38">
<note position="right" xlink:label="note-0107-03" xlink:href="note-0107-03a" xml:space="preserve">Fig. 134, <lb/>135, 136.</note>
</div>
<p>
  <s xml:id="echoid-s5802" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s5803" xml:space="preserve">E præmiſſo Theoremate non dificilè conficitur hoc _Problema:_ <lb/></s>
  <s xml:id="echoid-s5804" xml:space="preserve">Dato in axe puncto K, refractum deſignare, qui per hoc ipſum tran-<lb/>ſeat.</s>
  <s xml:id="echoid-s5805" xml:space="preserve">‖</s>
</p>
<p>
  <s xml:id="echoid-s5806" xml:space="preserve">Hoc nempe pacto. </s>
  <s xml:id="echoid-s5807" xml:space="preserve">Reperiatur punctum G, ut ſit KG. </s>
  <s xml:id="echoid-s5808" xml:space="preserve">AG :</s>
  <s xml:id="echoid-s5809" xml:space="preserve">: <lb/>
<anchor type="note" xlink:label="note-0107-04a" xlink:href="note-0107-04"/>
CK. </s>
  <s xml:id="echoid-s5810" xml:space="preserve">CR. </s>
  <s xml:id="echoid-s5811" xml:space="preserve">item fiat GF. </s>
  <s xml:id="echoid-s5812" xml:space="preserve">FA :</s>
  <s xml:id="echoid-s5813" xml:space="preserve">: CK. </s>
  <s xml:id="echoid-s5814" xml:space="preserve">CR (:</s>
  <s xml:id="echoid-s5815" xml:space="preserve">: KG. </s>
  <s xml:id="echoid-s5816" xml:space="preserve">AG). </s>
  <s xml:id="echoid-s5817" xml:space="preserve">tum <lb/>centro F, intervallo FG deſcribatur circulus refringentem interſecans <lb/>ad N; </s>
  <s xml:id="echoid-s5818" xml:space="preserve">erit connexa NK incidentis AN refractus.</s>
  <s xml:id="echoid-s5819" xml:space="preserve"/>
</p>
<div xml:id="echoid-div151" type="float" level="2" n="39">
<note position="right" xlink:label="note-0107-04" xlink:href="note-0107-04a" xml:space="preserve">Lect. 10. <lb/>Num. 25. <lb/>Fig. 137.</note>
</div>
<p>
  <s xml:id="echoid-s5820" xml:space="preserve">Nam ducatur FN; </s>
  <s xml:id="echoid-s5821" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5822" xml:space="preserve">ob KG. </s>
  <s xml:id="echoid-s5823" xml:space="preserve">AG :</s>
  <s xml:id="echoid-s5824" xml:space="preserve">: GF. </s>
  <s xml:id="echoid-s5825" xml:space="preserve">FA. </s>
  <s xml:id="echoid-s5826" xml:space="preserve">erit permu-<lb/>tatim KG. </s>
  <s xml:id="echoid-s5827" xml:space="preserve">GF :</s>
  <s xml:id="echoid-s5828" xml:space="preserve">: AG. </s>
  <s xml:id="echoid-s5829" xml:space="preserve">FA. </s>
  <s xml:id="echoid-s5830" xml:space="preserve">dividendóque KF. </s>
  <s xml:id="echoid-s5831" xml:space="preserve">GF :</s>
  <s xml:id="echoid-s5832" xml:space="preserve">: GF. </s>
  <s xml:id="echoid-s5833" xml:space="preserve">FA. <lb/></s>
  <s xml:id="echoid-s5834" xml:space="preserve">hoc eſt KF. </s>
  <s xml:id="echoid-s5835" xml:space="preserve">FN :</s>
  <s xml:id="echoid-s5836" xml:space="preserve">: FN. </s>
  <s xml:id="echoid-s5837" xml:space="preserve">FA. </s>
  <s xml:id="echoid-s5838" xml:space="preserve">quare triangula KFN, NFA
<pb o="90" file="0108" n="108" rhead=""/>
aſſimilantur. </s>
  <s xml:id="echoid-s5839" xml:space="preserve">unde NK. </s>
  <s xml:id="echoid-s5840" xml:space="preserve">KF :</s>
  <s xml:id="echoid-s5841" xml:space="preserve">: AN. </s>
  <s xml:id="echoid-s5842" xml:space="preserve">NF. </s>
  <s xml:id="echoid-s5843" xml:space="preserve">ſeu permutando NK. <lb/></s>
  <s xml:id="echoid-s5844" xml:space="preserve">AN :</s>
  <s xml:id="echoid-s5845" xml:space="preserve">: KF. </s>
  <s xml:id="echoid-s5846" xml:space="preserve">NF. </s>
  <s xml:id="echoid-s5847" xml:space="preserve">erat autem priùs KF. </s>
  <s xml:id="echoid-s5848" xml:space="preserve">NF :</s>
  <s xml:id="echoid-s5849" xml:space="preserve">: GF. </s>
  <s xml:id="echoid-s5850" xml:space="preserve">FA :</s>
  <s xml:id="echoid-s5851" xml:space="preserve">: CK. </s>
  <s xml:id="echoid-s5852" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0108-01a" xlink:href="note-0108-01"/>
CR. </s>
  <s xml:id="echoid-s5853" xml:space="preserve">eſt igitur NK. </s>
  <s xml:id="echoid-s5854" xml:space="preserve">AN :</s>
  <s xml:id="echoid-s5855" xml:space="preserve">: CK. </s>
  <s xml:id="echoid-s5856" xml:space="preserve">CR. </s>
  <s xml:id="echoid-s5857" xml:space="preserve">unde (juxta dictum Theo-<lb/>rema) conſtat factum.</s>
  <s xml:id="echoid-s5858" xml:space="preserve"/>
</p>
<div xml:id="echoid-div152" type="float" level="2" n="40">
<note position="left" xlink:label="note-0108-01" xlink:href="note-0108-01a" xml:space="preserve">Fig. 137.</note>
</div>
<p>
  <s xml:id="echoid-s5859" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s5860" xml:space="preserve">Ad conſtructionem iſtam advertentes animum, hujuſmodi <lb/>facilè _Conſectaria_ deducetis:</s>
  <s xml:id="echoid-s5861" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5862" xml:space="preserve">1. </s>
  <s xml:id="echoid-s5863" xml:space="preserve">Si circulus GNH_refringentem_ contingat ad H; </s>
  <s xml:id="echoid-s5864" xml:space="preserve">ipſius AH <lb/>(perpendicularis utique) refractus in K terminabitur; </s>
  <s xml:id="echoid-s5865" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5866" xml:space="preserve">aliorum <lb/>incidentium refracti ad unas ipſius K partes (ultra nempe vel citra <lb/>K reſpectu centri, pro diverſitate caſuum ab ipſius A poſitione reſul-<lb/>tantium) cadent.</s>
  <s xml:id="echoid-s5867" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5868" xml:space="preserve">2. </s>
  <s xml:id="echoid-s5869" xml:space="preserve">Si dictus ille circulus _refringenti_ non occurrat omninò, _Problema_ <lb/>conſtructionem reſpuet; </s>
  <s xml:id="echoid-s5870" xml:space="preserve">nec ullus refractus punctum K permeabit.</s>
  <s xml:id="echoid-s5871" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5872" xml:space="preserve">3. </s>
  <s xml:id="echoid-s5873" xml:space="preserve">Si circulus GNH_refringenti_ coincidat (id quod facilè concipi <lb/>poteſt, &amp; </s>
  <s xml:id="echoid-s5874" xml:space="preserve">in aliquo reverà caſu contingit) omnes refracti in punctum <lb/>K confluent.</s>
  <s xml:id="echoid-s5875" xml:space="preserve">‖ Hæc &amp; </s>
  <s xml:id="echoid-s5876" xml:space="preserve">alia conſtructionem iſtam conſectantur ſoler-<lb/>ter expanſem; </s>
  <s xml:id="echoid-s5877" xml:space="preserve">quorum ſaltem nonnulla haud abs re fuerit exertiùs <lb/>oſtendi; </s>
  <s xml:id="echoid-s5878" xml:space="preserve">velut hoc imprimìs palmarium.</s>
  <s xml:id="echoid-s5879" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5880" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s5881" xml:space="preserve">Si fuerit AB. </s>
  <s xml:id="echoid-s5882" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s5883" xml:space="preserve">: BZ. </s>
  <s xml:id="echoid-s5884" xml:space="preserve">CZ; </s>
  <s xml:id="echoid-s5885" xml:space="preserve">dico punctum Z eſſe limi-<lb/>tem, ultra vel citra quem nullus refractus axim interſecat; </s>
  <s xml:id="echoid-s5886" xml:space="preserve">ſeu per-<lb/>pendicularis ipſius AB refractum in Z terminari.</s>
  <s xml:id="echoid-s5887" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5888" xml:space="preserve">Nam cujuſvis incidentis AN refractus axi occurrat in K, erit ideò <lb/>CK. </s>
  <s xml:id="echoid-s5889" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s5890" xml:space="preserve">: NK. </s>
  <s xml:id="echoid-s5891" xml:space="preserve">NA. </s>
  <s xml:id="echoid-s5892" xml:space="preserve">ergò quum ſit CR. </s>
  <s xml:id="echoid-s5893" xml:space="preserve">CZ :</s>
  <s xml:id="echoid-s5894" xml:space="preserve">: AB. </s>
  <s xml:id="echoid-s5895" xml:space="preserve">BZ; <lb/></s>
  <s xml:id="echoid-s5896" xml:space="preserve">erit CK. </s>
  <s xml:id="echoid-s5897" xml:space="preserve">CR + CR. </s>
  <s xml:id="echoid-s5898" xml:space="preserve">CZ = NK. </s>
  <s xml:id="echoid-s5899" xml:space="preserve">NA + AB. </s>
  <s xml:id="echoid-s5900" xml:space="preserve">BZ.</s>
  <s xml:id="echoid-s5901" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s5902" xml:space="preserve">1. </s>
  <s xml:id="echoid-s5903" xml:space="preserve">Eſt autem (in prima figura, ubi puncta Z, &amp; </s>
  <s xml:id="echoid-s5904" xml:space="preserve">K ſunt ad partes <lb/>
<anchor type="note" xlink:label="note-0108-02a" xlink:href="note-0108-02"/>
centri, vel ubi refracti ad axem directè procurrentes convergunt) <lb/>BK&amp;</s>
  <s xml:id="echoid-s5905" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5906" xml:space="preserve">NK, &amp; </s>
  <s xml:id="echoid-s5907" xml:space="preserve">AB&amp;</s>
  <s xml:id="echoid-s5908" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5909" xml:space="preserve">AN; </s>
  <s xml:id="echoid-s5910" xml:space="preserve">adeóque BK. </s>
  <s xml:id="echoid-s5911" xml:space="preserve">AB &amp;</s>
  <s xml:id="echoid-s5912" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5913" xml:space="preserve">NK. </s>
  <s xml:id="echoid-s5914" xml:space="preserve">NA. <lb/></s>
  <s xml:id="echoid-s5915" xml:space="preserve">ergò CK. </s>
  <s xml:id="echoid-s5916" xml:space="preserve">CR + CR. </s>
  <s xml:id="echoid-s5917" xml:space="preserve">CZ&amp;</s>
  <s xml:id="echoid-s5918" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5919" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s5920" xml:space="preserve">AB + AB. </s>
  <s xml:id="echoid-s5921" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s5922" xml:space="preserve">hoc eſt <lb/>C K. </s>
  <s xml:id="echoid-s5923" xml:space="preserve">CZ&amp;</s>
  <s xml:id="echoid-s5924" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5925" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s5926" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s5927" xml:space="preserve">vel inversè permutando BK. </s>
  <s xml:id="echoid-s5928" xml:space="preserve">CK&amp;</s>
  <s xml:id="echoid-s5929" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5930" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s5931" xml:space="preserve"><lb/>C Z. </s>
  <s xml:id="echoid-s5932" xml:space="preserve">dividendóque BC. </s>
  <s xml:id="echoid-s5933" xml:space="preserve">CK&amp;</s>
  <s xml:id="echoid-s5934" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5935" xml:space="preserve">BC. </s>
  <s xml:id="echoid-s5936" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s5937" xml:space="preserve">ergò CK &amp;</s>
  <s xml:id="echoid-s5938" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5939" xml:space="preserve">CZ; </s>
  <s xml:id="echoid-s5940" xml:space="preserve"><lb/>adeóque punctum K ſupra Z exiſtit, verſus centrum; </s>
  <s xml:id="echoid-s5941" xml:space="preserve">quod erat pro-<lb/>poſitum oſtendere.</s>
  <s xml:id="echoid-s5942" xml:space="preserve"/>
</p>
<div xml:id="echoid-div153" type="float" level="2" n="41">
<note position="left" xlink:label="note-0108-02" xlink:href="note-0108-02a" xml:space="preserve">Fig. 138.</note>
</div>
<p>
  <s xml:id="echoid-s5943" xml:space="preserve">2. </s>
  <s xml:id="echoid-s5944" xml:space="preserve">In ſecundâ verò figurá ubi puncta Z, K ad alteras ſupra punctum <lb/>
<anchor type="note" xlink:label="note-0108-03a" xlink:href="note-0108-03"/>
A partes à centro averſas cadunt) connectatur ſubtenſa BN, &amp; </s>
  <s xml:id="echoid-s5945" xml:space="preserve">du-<lb/>catur AS ad KN parallela; </s>
  <s xml:id="echoid-s5946" xml:space="preserve">hæc ſecabit angulum BA N, majorem <lb/>ipſo BK N, vel BA S; </s>
  <s xml:id="echoid-s5947" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s5948" xml:space="preserve">cùm angulus ABNfit obtuſus, èrit AN <lb/>&amp;</s>
  <s xml:id="echoid-s5949" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5950" xml:space="preserve">AS. </s>
  <s xml:id="echoid-s5951" xml:space="preserve">adeóque KN. </s>
  <s xml:id="echoid-s5952" xml:space="preserve">AN&amp;</s>
  <s xml:id="echoid-s5953" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5954" xml:space="preserve">KN. </s>
  <s xml:id="echoid-s5955" xml:space="preserve">AS :</s>
  <s xml:id="echoid-s5956" xml:space="preserve">: KB. </s>
  <s xml:id="echoid-s5957" xml:space="preserve">AB. </s>
  <s xml:id="echoid-s5958" xml:space="preserve">erit etiam hîc <lb/>igitur (ut ſupra)C K. </s>
  <s xml:id="echoid-s5959" xml:space="preserve">CZ&amp;</s>
  <s xml:id="echoid-s5960" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5961" xml:space="preserve">BK. </s>
  <s xml:id="echoid-s5962" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s5963" xml:space="preserve">vel permutatim CK. </s>
  <s xml:id="echoid-s5964" xml:space="preserve">BK
<pb o="91" file="0109" n="109" rhead=""/>
&amp;</s>
  <s xml:id="echoid-s5965" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5966" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s5967" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s5968" xml:space="preserve">dividendóque CB. </s>
  <s xml:id="echoid-s5969" xml:space="preserve">BK&amp;</s>
  <s xml:id="echoid-s5970" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5971" xml:space="preserve">CB. </s>
  <s xml:id="echoid-s5972" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s5973" xml:space="preserve">adeóque BK <lb/>&amp;</s>
  <s xml:id="echoid-s5974" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5975" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s5976" xml:space="preserve">hoc eſt punctum K magìs quàm Z à centro elongatur.</s>
  <s xml:id="echoid-s5977" xml:space="preserve"/>
</p>
<div xml:id="echoid-div154" type="float" level="2" n="42">
<note position="left" xlink:label="note-0108-03" xlink:href="note-0108-03a" xml:space="preserve">Fig. 139.</note>
</div>
<p>
  <s xml:id="echoid-s5978" xml:space="preserve">3. </s>
  <s xml:id="echoid-s5979" xml:space="preserve">Haud diſſimilis in aliis caſibus erìt _Demonſtratio_; </s>
  <s xml:id="echoid-s5980" xml:space="preserve">ut in hoc, ubi <lb/>
<anchor type="note" xlink:label="note-0109-01a" xlink:href="note-0109-01"/>
I &amp;</s>
  <s xml:id="echoid-s5981" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s5982" xml:space="preserve">R, ad convexas; </s>
  <s xml:id="echoid-s5983" xml:space="preserve">eſt enim hîc (ut in præcedente) KB. </s>
  <s xml:id="echoid-s5984" xml:space="preserve">AB&amp;</s>
  <s xml:id="echoid-s5985" xml:space="preserve">lt; <lb/></s>
  <s xml:id="echoid-s5986" xml:space="preserve">KN. </s>
  <s xml:id="echoid-s5987" xml:space="preserve">AN. </s>
  <s xml:id="echoid-s5988" xml:space="preserve">adeóque (ſupra monſtratis inſiſtendo) CK.</s>
  <s xml:id="echoid-s5989" xml:space="preserve">CZ&amp;</s>
  <s xml:id="echoid-s5990" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5991" xml:space="preserve"><lb/>KB. </s>
  <s xml:id="echoid-s5992" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s5993" xml:space="preserve">vel permutando CK. </s>
  <s xml:id="echoid-s5994" xml:space="preserve">KB&amp;</s>
  <s xml:id="echoid-s5995" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s5996" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s5997" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s5998" xml:space="preserve">dividendóque <lb/>CB. </s>
  <s xml:id="echoid-s5999" xml:space="preserve">KB&amp;</s>
  <s xml:id="echoid-s6000" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6001" xml:space="preserve">CB. </s>
  <s xml:id="echoid-s6002" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s6003" xml:space="preserve">unde KB&amp;</s>
  <s xml:id="echoid-s6004" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6005" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s6006" xml:space="preserve">adeóque punctum K <lb/>centro ſemper vicinius eſt quàm Z.</s>
  <s xml:id="echoid-s6007" xml:space="preserve"/>
</p>
<div xml:id="echoid-div155" type="float" level="2" n="43">
<note position="right" xlink:label="note-0109-01" xlink:href="note-0109-01a" xml:space="preserve">Fig. 140.</note>
</div>
<p>
  <s xml:id="echoid-s6008" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s6009" xml:space="preserve">Hæc autem cùm, modo ſuo mutatis mutandis, ad omnes <lb/>caſus transferri poſſint, habentur indè determinati refractorum limites, <lb/>hoc eſt apparentia radiantium punctorum A loca, reſpectu oculi cen-<lb/>trum habentis in axe AC ſitum; </s>
  <s xml:id="echoid-s6010" xml:space="preserve">juxta doctrinam à nobis toties in-<lb/>culcatam.</s>
  <s xml:id="echoid-s6011" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6012" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s6013" xml:space="preserve">Id autem hîc in duobus caſibus (utroque nimirum ad circuli <lb/>cavas) peculiare venit obſervandum cùm ſit CB = CR, omnes <lb/>refractos in ipſo puncto Z (ut ſuprà definito) retrò protractos con-<lb/>gregari. </s>
  <s xml:id="echoid-s6014" xml:space="preserve">Nam ob AB. </s>
  <s xml:id="echoid-s6015" xml:space="preserve">BC :</s>
  <s xml:id="echoid-s6016" xml:space="preserve">: AB. </s>
  <s xml:id="echoid-s6017" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s6018" xml:space="preserve">: BZ. </s>
  <s xml:id="echoid-s6019" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s6020" xml:space="preserve">erit divi-<lb/>dendo AC. </s>
  <s xml:id="echoid-s6021" xml:space="preserve">BC :</s>
  <s xml:id="echoid-s6022" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6023" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s6024" xml:space="preserve">quapropter ad punctum quodvis N <lb/>adſumptum connexis AN, ZN, erit ZN. </s>
  <s xml:id="echoid-s6025" xml:space="preserve">AN :</s>
  <s xml:id="echoid-s6026" xml:space="preserve">: (CZ. </s>
  <s xml:id="echoid-s6027" xml:space="preserve">CN :</s>
  <s xml:id="echoid-s6028" xml:space="preserve">: ) <lb/>CZ. </s>
  <s xml:id="echoid-s6029" xml:space="preserve">CR. </s>
  <s xml:id="echoid-s6030" xml:space="preserve">unde ZN refractus erit incidentis AN.</s>
  <s xml:id="echoid-s6031" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6032" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s6033" xml:space="preserve">Hinc etiam ſi fuerit AB = CR, conſequetur punctum Z <lb/>
<anchor type="note" xlink:label="note-0109-02a" xlink:href="note-0109-02"/>
à centro infinitè diſtare; </s>
  <s xml:id="echoid-s6034" xml:space="preserve">quia nempe tum ob AB. </s>
  <s xml:id="echoid-s6035" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s6036" xml:space="preserve">: BZ. </s>
  <s xml:id="echoid-s6037" xml:space="preserve">CZ, <lb/>erit BZ = CZ; </s>
  <s xml:id="echoid-s6038" xml:space="preserve">id quod fieri nequit, niſi punctum Z ità elongetur <lb/>infinitè.</s>
  <s xml:id="echoid-s6039" xml:space="preserve"/>
</p>
<div xml:id="echoid-div156" type="float" level="2" n="44">
<note position="right" xlink:label="note-0109-02" xlink:href="note-0109-02a" xml:space="preserve">Fig. 141.</note>
</div>
<p>
  <s xml:id="echoid-s6040" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s6041" xml:space="preserve">_Conſectantur_ &amp; </s>
  <s xml:id="echoid-s6042" xml:space="preserve">hæc: </s>
  <s xml:id="echoid-s6043" xml:space="preserve">Si punctorum radiantium A, _a_ limites <lb/>
<anchor type="note" xlink:label="note-0109-03a" xlink:href="note-0109-03"/>
ſint puncta Z, ζ, erit AC. </s>
  <s xml:id="echoid-s6044" xml:space="preserve">AB + BZ. </s>
  <s xml:id="echoid-s6045" xml:space="preserve">CZ = _a_ C. </s>
  <s xml:id="echoid-s6046" xml:space="preserve">_a_B + Bζ. <lb/></s>
  <s xml:id="echoid-s6047" xml:space="preserve">C ζ.</s>
  <s xml:id="echoid-s6048" xml:space="preserve"/>
</p>
<div xml:id="echoid-div157" type="float" level="2" n="45">
<note position="right" xlink:label="note-0109-03" xlink:href="note-0109-03a" xml:space="preserve">Fig. 142, <lb/>143.</note>
</div>
<p>
  <s xml:id="echoid-s6049" xml:space="preserve">Nam è præmiſſis facilè conſtat eſſe <lb/>tam AC. </s>
  <s xml:id="echoid-s6050" xml:space="preserve">AB + BZ. </s>
  <s xml:id="echoid-s6051" xml:space="preserve">CZ = \q̇uam _a_ C. </s>
  <s xml:id="echoid-s6052" xml:space="preserve">_a_ B + B ζ. </s>
  <s xml:id="echoid-s6053" xml:space="preserve">C ζ = }I. </s>
  <s xml:id="echoid-s6054" xml:space="preserve">R.</s>
  <s xml:id="echoid-s6055" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6056" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s6057" xml:space="preserve">Unde Cζ &amp;</s>
  <s xml:id="echoid-s6058" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6059" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s6060" xml:space="preserve">Nam ob BC. </s>
  <s xml:id="echoid-s6061" xml:space="preserve">AB &amp;</s>
  <s xml:id="echoid-s6062" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6063" xml:space="preserve">BC. </s>
  <s xml:id="echoid-s6064" xml:space="preserve">_a_B. <lb/></s>
  <s xml:id="echoid-s6065" xml:space="preserve">componendóque AC. </s>
  <s xml:id="echoid-s6066" xml:space="preserve">AB &amp;</s>
  <s xml:id="echoid-s6067" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6068" xml:space="preserve">_a_ C. </s>
  <s xml:id="echoid-s6069" xml:space="preserve">_a_ B. </s>
  <s xml:id="echoid-s6070" xml:space="preserve">erit BZ. </s>
  <s xml:id="echoid-s6071" xml:space="preserve">CZ &amp;</s>
  <s xml:id="echoid-s6072" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6073" xml:space="preserve">BC α B. </s>
  <s xml:id="echoid-s6074" xml:space="preserve"><lb/>Cζ. </s>
  <s xml:id="echoid-s6075" xml:space="preserve">dividendóque BC. </s>
  <s xml:id="echoid-s6076" xml:space="preserve">CZ &amp;</s>
  <s xml:id="echoid-s6077" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6078" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s6079" xml:space="preserve">Cζ. </s>
  <s xml:id="echoid-s6080" xml:space="preserve">adeóque Cζ &amp;</s>
  <s xml:id="echoid-s6081" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6082" xml:space="preserve"><lb/>C Z.</s>
  <s xml:id="echoid-s6083" xml:space="preserve"/>
</p>
<pb o="92" file="0110" n="110" rhead=""/>
<p>
  <s xml:id="echoid-s6084" xml:space="preserve">XXI. </s>
  <s xml:id="echoid-s6085" xml:space="preserve">Imò univerſim ſi radii quivis AF, _a_ φ ad circulum refrin-<lb/>
<anchor type="note" xlink:label="note-0110-01a" xlink:href="note-0110-01"/>
gentem æqualiter inclinentur, híſque conveniant refracti FL, φ λ, <lb/>erit C λ &amp;</s>
  <s xml:id="echoid-s6086" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6087" xml:space="preserve">CL. </s>
  <s xml:id="echoid-s6088" xml:space="preserve">id quod hoc modo non inelegantèr oſtenditur. </s>
  <s xml:id="echoid-s6089" xml:space="preserve">Du-<lb/>catur recta BX cum BC angulum efficiens parem angulo refracto <lb/>ad poſitam inclinationem pertinenti; </s>
  <s xml:id="echoid-s6090" xml:space="preserve">perque puncta F, φ; </s>
  <s xml:id="echoid-s6091" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6092" xml:space="preserve">cen-<lb/>trum C tranſeuntes rectæ ipſi BX occurant punctis P, π. </s>
  <s xml:id="echoid-s6093" xml:space="preserve">tum quo-<lb/>niam triangula FC L, BCPæquiangula ſunt (angulus enim CB P <lb/>angulo CFLex conſtructione par eſt, &amp; </s>
  <s xml:id="echoid-s6094" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s6095" xml:space="preserve">BCPverticali ſuo <lb/>FCLæquatur) nec non latus CB lateri CF æquatur, erit CP = <lb/>C L. </s>
  <s xml:id="echoid-s6096" xml:space="preserve">Simili planè diſcurſu eſt C π = C λ. </s>
  <s xml:id="echoid-s6097" xml:space="preserve">Porrò quia C φ ad <lb/>C _a_ (hoc eſt Sinus anguli C _a_ φ ad Sinum anguli C φ _a_) majorem <lb/>rationem habet, quàm CF ad CA (hoc eſt quàm Sinus anguli CA F <lb/>ad Sinum anguli AF C, vel æ qualis anguli C φ α) liquet angulum <lb/>C _a_ φ majorem eſſe angulo CA F, adeóque reliquum _a_ C φ minorem <lb/>eſſe reliquo AC F; </s>
  <s xml:id="echoid-s6098" xml:space="preserve">vel angulum PCBangulo π CB. </s>
  <s xml:id="echoid-s6099" xml:space="preserve">unde liquet <lb/>eſſe C π majorem quàm CP; </s>
  <s xml:id="echoid-s6100" xml:space="preserve">hoc eſt C λ majorem eſſe quàm CL: <lb/></s>
  <s xml:id="echoid-s6101" xml:space="preserve">Quod E. </s>
  <s xml:id="echoid-s6102" xml:space="preserve">D.</s>
  <s xml:id="echoid-s6103" xml:space="preserve"/>
</p>
<div xml:id="echoid-div158" type="float" level="2" n="46">
<note position="left" xlink:label="note-0110-01" xlink:href="note-0110-01a" xml:space="preserve">Fig. 144.</note>
</div>
<p>
  <s xml:id="echoid-s6104" xml:space="preserve">_Coroll._ </s>
  <s xml:id="echoid-s6105" xml:space="preserve">Vides arcum BF majorem eſſe arcu B φ.</s>
  <s xml:id="echoid-s6106" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6107" xml:space="preserve">Notes etiam omnes ejuſdem inclinationis refractos ope ductæ rectæ <lb/>BX promptiſſimè deſignari. </s>
  <s xml:id="echoid-s6108" xml:space="preserve">ſed hæc an πρργδ fuerint neſcio.</s>
  <s xml:id="echoid-s6109" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6110" xml:space="preserve">XXII. </s>
  <s xml:id="echoid-s6111" xml:space="preserve">_Subjiciam &amp; </s>
  <s xml:id="echoid-s6112" xml:space="preserve">hoc Theorema:_ </s>
  <s xml:id="echoid-s6113" xml:space="preserve">Convexo denſiori inciden-<lb/>
<anchor type="note" xlink:label="note-0110-02a" xlink:href="note-0110-02"/>
tiùm radiorum AM, AN (quorum AN ſit obliquior) refracti <lb/>MK, NL axem ad eaſdem partes, directè pergentes, ſecent, iſte ad K, <lb/>hic ad L; </s>
  <s xml:id="echoid-s6114" xml:space="preserve">dico fore CK majorem quàm CL.</s>
  <s xml:id="echoid-s6115" xml:space="preserve"/>
</p>
<div xml:id="echoid-div159" type="float" level="2" n="47">
<note position="left" xlink:label="note-0110-02" xlink:href="note-0110-02a" xml:space="preserve">Fig. 145.</note>
</div>
<p>
  <s xml:id="echoid-s6116" xml:space="preserve">Nam connexis CN, KN; </s>
  <s xml:id="echoid-s6117" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6118" xml:space="preserve">ductâ LH ad KN parallelâ quo-<lb/>niam, è præmiſſis, eſt CK. </s>
  <s xml:id="echoid-s6119" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s6120" xml:space="preserve">: MK. </s>
  <s xml:id="echoid-s6121" xml:space="preserve">MA. </s>
  <s xml:id="echoid-s6122" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6123" xml:space="preserve">CR. </s>
  <s xml:id="echoid-s6124" xml:space="preserve">CL :</s>
  <s xml:id="echoid-s6125" xml:space="preserve">: <lb/>NA. </s>
  <s xml:id="echoid-s6126" xml:space="preserve">NL. </s>
  <s xml:id="echoid-s6127" xml:space="preserve">erit CK. </s>
  <s xml:id="echoid-s6128" xml:space="preserve">CK + CR. </s>
  <s xml:id="echoid-s6129" xml:space="preserve">CL = MK. </s>
  <s xml:id="echoid-s6130" xml:space="preserve">MA + NA. </s>
  <s xml:id="echoid-s6131" xml:space="preserve">NL. <lb/></s>
  <s xml:id="echoid-s6132" xml:space="preserve">eſt autem NK. </s>
  <s xml:id="echoid-s6133" xml:space="preserve">NA&amp;</s>
  <s xml:id="echoid-s6134" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6135" xml:space="preserve">MK. </s>
  <s xml:id="echoid-s6136" xml:space="preserve">MA (quia NK&amp;</s>
  <s xml:id="echoid-s6137" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6138" xml:space="preserve">MK, &amp; </s>
  <s xml:id="echoid-s6139" xml:space="preserve">NA <lb/>&amp;</s>
  <s xml:id="echoid-s6140" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6141" xml:space="preserve">MA)ergo CK. </s>
  <s xml:id="echoid-s6142" xml:space="preserve">CR + CR. </s>
  <s xml:id="echoid-s6143" xml:space="preserve">CL&amp;</s>
  <s xml:id="echoid-s6144" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6145" xml:space="preserve">NK. </s>
  <s xml:id="echoid-s6146" xml:space="preserve">NA + NA. </s>
  <s xml:id="echoid-s6147" xml:space="preserve"><lb/>NL. </s>
  <s xml:id="echoid-s6148" xml:space="preserve">hoc eſt CK. </s>
  <s xml:id="echoid-s6149" xml:space="preserve">CL &amp;</s>
  <s xml:id="echoid-s6150" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6151" xml:space="preserve">NK. </s>
  <s xml:id="echoid-s6152" xml:space="preserve">NL. </s>
  <s xml:id="echoid-s6153" xml:space="preserve">hoc eſt NK. </s>
  <s xml:id="echoid-s6154" xml:space="preserve">HL. </s>
  <s xml:id="echoid-s6155" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s6156" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6157" xml:space="preserve"><lb/>NK. </s>
  <s xml:id="echoid-s6158" xml:space="preserve">NL. </s>
  <s xml:id="echoid-s6159" xml:space="preserve">quapropter eſt LH&amp;</s>
  <s xml:id="echoid-s6160" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6161" xml:space="preserve">NL. </s>
  <s xml:id="echoid-s6162" xml:space="preserve">eſt autem angulus LCN <lb/>obtufus; </s>
  <s xml:id="echoid-s6163" xml:space="preserve">ergò recta LH angulum CLNſecat; </s>
  <s xml:id="echoid-s6164" xml:space="preserve">ac angulus LHC <lb/>interno LNCmajor eſt; </s>
  <s xml:id="echoid-s6165" xml:space="preserve">hoc eſt angulus KNCangulo LNC <lb/>major eſt. </s>
  <s xml:id="echoid-s6166" xml:space="preserve">unde liquidò patet fore CK&amp;</s>
  <s xml:id="echoid-s6167" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6168" xml:space="preserve">CL.</s>
  <s xml:id="echoid-s6169" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6170" xml:space="preserve">_Coroll._ </s>
  <s xml:id="echoid-s6171" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s6172" xml:space="preserve">CL = MK. </s>
  <s xml:id="echoid-s6173" xml:space="preserve">MA + NA. </s>
  <s xml:id="echoid-s6174" xml:space="preserve">NL.</s>
  <s xml:id="echoid-s6175" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6176" xml:space="preserve">XXIII. </s>
  <s xml:id="echoid-s6177" xml:space="preserve">Hinc, ejuſmodi omnes refracti ſeipſos priùs quàm axem <lb/>interſecant, velut ad X.</s>
  <s xml:id="echoid-s6178" xml:space="preserve">‖ Hoc ſpeciminis loco pro caſu, qui præ
<pb o="93" file="0111" n="111" rhead=""/>
manibus. </s>
  <s xml:id="echoid-s6179" xml:space="preserve">propter alios qui ſimilia volet, ipſeviderit, &amp; </s>
  <s xml:id="echoid-s6180" xml:space="preserve">ſibi para-<lb/>verit. </s>
  <s xml:id="echoid-s6181" xml:space="preserve">ego jam aliò progredior; </s>
  <s xml:id="echoid-s6182" xml:space="preserve">eò ſcilicet, ut locum definiam <lb/>imaginis in dato quovis refracto apparentis; </s>
  <s xml:id="echoid-s6183" xml:space="preserve">prætervehemur enim <lb/>illud in his certè caſibus _intricatiſſimum Problema_ (cujúſque Solutio <lb/>nullatenus aut laborem quem exigit, aut temporis jacturam compen-<lb/>ſabit) quo jubetur per datum punctum tranſeuntem refractum deſig-<lb/>nare. </s>
  <s xml:id="echoid-s6184" xml:space="preserve">poſitione datum igitur refractum accipimus; </s>
  <s xml:id="echoid-s6185" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6186" xml:space="preserve">in hoc ima-<lb/>ginis locum ex hoc uno Theoremate determinamus.</s>
  <s xml:id="echoid-s6187" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6188" xml:space="preserve">XXIV. </s>
  <s xml:id="echoid-s6189" xml:space="preserve">Duorum incidentium ANP, ARS ſibi quàm proximo-<lb/>
<anchor type="note" xlink:label="note-0111-01a" xlink:href="note-0111-01"/>
rum concipiantur refracti N π, R σ ſeſe puncto Z decuſſantes; </s>
  <s xml:id="echoid-s6190" xml:space="preserve">biſe-<lb/>centúrque ſubtenſæ NP, N π punctis E, G; </s>
  <s xml:id="echoid-s6191" xml:space="preserve">(à rectis nempe CE, <lb/>CG ad illas perpendicularibus) dico rationem NZ ad GZ è ratio-<lb/>nibus CE ad CG (hoc eſt 1. </s>
  <s xml:id="echoid-s6192" xml:space="preserve">R), NG ad NE, ac AN ad AE <lb/>componi.</s>
  <s xml:id="echoid-s6193" xml:space="preserve"/>
</p>
<div xml:id="echoid-div160" type="float" level="2" n="48">
<note position="right" xlink:label="note-0111-01" xlink:href="note-0111-01a" xml:space="preserve">Fig. 146.</note>
</div>
<p>
  <s xml:id="echoid-s6194" xml:space="preserve">Ducantur enim CK ad RS, &amp; </s>
  <s xml:id="echoid-s6195" xml:space="preserve">CI ad R σ perpendiculares; </s>
  <s xml:id="echoid-s6196" xml:space="preserve">in <lb/>que producetis CE, CG capiantur CF = CK; </s>
  <s xml:id="echoid-s6197" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6198" xml:space="preserve">CH = CI; <lb/></s>
  <s xml:id="echoid-s6199" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6200" xml:space="preserve">per F ducatur TV ad NP parallela; </s>
  <s xml:id="echoid-s6201" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6202" xml:space="preserve">per H etiam XY ad N π <lb/>parallela. </s>
  <s xml:id="echoid-s6203" xml:space="preserve">Jam eſt AP. </s>
  <s xml:id="echoid-s6204" xml:space="preserve">AN :</s>
  <s xml:id="echoid-s6205" xml:space="preserve">: arc PS. </s>
  <s xml:id="echoid-s6206" xml:space="preserve">arc NR (ob ſumptam <lb/>arcuum indefinitam parvitatem). </s>
  <s xml:id="echoid-s6207" xml:space="preserve">ergò {AP ±: </s>
  <s xml:id="echoid-s6208" xml:space="preserve">AN/2}. </s>
  <s xml:id="echoid-s6209" xml:space="preserve">AN :</s>
  <s xml:id="echoid-s6210" xml:space="preserve">: <lb/>{arc PS ±: </s>
  <s xml:id="echoid-s6211" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s6212" xml:space="preserve">NR/2}. </s>
  <s xml:id="echoid-s6213" xml:space="preserve">arc NR. </s>
  <s xml:id="echoid-s6214" xml:space="preserve">hoc eſt AE. </s>
  <s xml:id="echoid-s6215" xml:space="preserve">AN :</s>
  <s xml:id="echoid-s6216" xml:space="preserve">: arc NT. </s>
  <s xml:id="echoid-s6217" xml:space="preserve">arc <lb/>NR. </s>
  <s xml:id="echoid-s6218" xml:space="preserve">item eſt NZ. </s>
  <s xml:id="echoid-s6219" xml:space="preserve">Z π :</s>
  <s xml:id="echoid-s6220" xml:space="preserve">: arc NR. </s>
  <s xml:id="echoid-s6221" xml:space="preserve">arc πσ. </s>
  <s xml:id="echoid-s6222" xml:space="preserve">ac indè NZ. </s>
  <s xml:id="echoid-s6223" xml:space="preserve"><lb/>{NZ ±: </s>
  <s xml:id="echoid-s6224" xml:space="preserve">Z π/2} :</s>
  <s xml:id="echoid-s6225" xml:space="preserve">: arc NR. </s>
  <s xml:id="echoid-s6226" xml:space="preserve">{arc NR ±: </s>
  <s xml:id="echoid-s6227" xml:space="preserve">πσ/2}. </s>
  <s xml:id="echoid-s6228" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/>hoc eſt NZ. </s>
  <s xml:id="echoid-s6229" xml:space="preserve">ZG :</s>
  <s xml:id="echoid-s6230" xml:space="preserve">: arc
<anchor type="note" xlink:label="note-0111-02a" xlink:href="note-0111-02"/>
NR. </s>
  <s xml:id="echoid-s6231" xml:space="preserve">arc NX. </s>
  <s xml:id="echoid-s6232" xml:space="preserve">ergò, rationes æquales adjungendo, eſt. </s>
  <s xml:id="echoid-s6233" xml:space="preserve">AE. </s>
  <s xml:id="echoid-s6234" xml:space="preserve">AN <lb/>+ NZ. </s>
  <s xml:id="echoid-s6235" xml:space="preserve">ZG = arc NT. </s>
  <s xml:id="echoid-s6236" xml:space="preserve">arc NR + arc NR. </s>
  <s xml:id="echoid-s6237" xml:space="preserve">arc NX = arc <lb/>NT. </s>
  <s xml:id="echoid-s6238" xml:space="preserve">arc NX. </s>
  <s xml:id="echoid-s6239" xml:space="preserve">quoniam autem eſt CE. </s>
  <s xml:id="echoid-s6240" xml:space="preserve">CG :</s>
  <s xml:id="echoid-s6241" xml:space="preserve">: (I. </s>
  <s xml:id="echoid-s6242" xml:space="preserve">R :</s>
  <s xml:id="echoid-s6243" xml:space="preserve">: CK. <lb/></s>
  <s xml:id="echoid-s6244" xml:space="preserve">CI :</s>
  <s xml:id="echoid-s6245" xml:space="preserve">:) CF. </s>
  <s xml:id="echoid-s6246" xml:space="preserve">CH. </s>
  <s xml:id="echoid-s6247" xml:space="preserve">vel permutando CE. </s>
  <s xml:id="echoid-s6248" xml:space="preserve">CF :</s>
  <s xml:id="echoid-s6249" xml:space="preserve">: CG. </s>
  <s xml:id="echoid-s6250" xml:space="preserve">CH; </s>
  <s xml:id="echoid-s6251" xml:space="preserve">erit, <lb/>
<anchor type="note" xlink:href="" symbol="*"/>juxta præmonſtrata, arc NT. </s>
  <s xml:id="echoid-s6252" xml:space="preserve">arc NX = NG. </s>
  <s xml:id="echoid-s6253" xml:space="preserve">NE + CE. </s>
  <s xml:id="echoid-s6254" xml:space="preserve">CG.</s>
  <s xml:id="echoid-s6255" xml:space="preserve">
<anchor type="note" xlink:label="note-0111-03a" xlink:href="note-0111-03"/>
quapropter erit AE. </s>
  <s xml:id="echoid-s6256" xml:space="preserve">AN + NZ. </s>
  <s xml:id="echoid-s6257" xml:space="preserve">ZG = NG. </s>
  <s xml:id="echoid-s6258" xml:space="preserve">NF + CE. <lb/></s>
  <s xml:id="echoid-s6259" xml:space="preserve">CG. </s>
  <s xml:id="echoid-s6260" xml:space="preserve">unde (rationes hinc indè pares ſubducendo) erit NZ. </s>
  <s xml:id="echoid-s6261" xml:space="preserve">ZG :</s>
  <s xml:id="echoid-s6262" xml:space="preserve">: <lb/>+ CE. </s>
  <s xml:id="echoid-s6263" xml:space="preserve">CG + NG. </s>
  <s xml:id="echoid-s6264" xml:space="preserve">NE + AN. </s>
  <s xml:id="echoid-s6265" xml:space="preserve">AE. </s>
  <s xml:id="echoid-s6266" xml:space="preserve">Quod propoſitum fuit <lb/>oſtendere.</s>
  <s xml:id="echoid-s6267" xml:space="preserve"/>
</p>
<div xml:id="echoid-div161" type="float" level="2" n="49">
<note symbol="*" position="right" xlink:label="note-0111-02" xlink:href="note-0111-02a" xml:space="preserve">_Lect. 9._ <lb/>_Num. iI_</note>
<note symbol="*" position="right" xlink:label="note-0111-03" xlink:href="note-0111-03a" xml:space="preserve">_12 Lect._ <lb/>_Num. 6._</note>
</div>
<p>
  <s xml:id="echoid-s6268" xml:space="preserve">XXV. </s>
  <s xml:id="echoid-s6269" xml:space="preserve">Hinc, ſi fiat CE. </s>
  <s xml:id="echoid-s6270" xml:space="preserve">CG :</s>
  <s xml:id="echoid-s6271" xml:space="preserve">: NE. </s>
  <s xml:id="echoid-s6272" xml:space="preserve">L; </s>
  <s xml:id="echoid-s6273" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6274" xml:space="preserve">AN. </s>
  <s xml:id="echoid-s6275" xml:space="preserve">AE :</s>
  <s xml:id="echoid-s6276" xml:space="preserve">: L. <lb/></s>
  <s xml:id="echoid-s6277" xml:space="preserve">M; </s>
  <s xml:id="echoid-s6278" xml:space="preserve">erit NZ ZG :</s>
  <s xml:id="echoid-s6279" xml:space="preserve">: NG. </s>
  <s xml:id="echoid-s6280" xml:space="preserve">M. </s>
  <s xml:id="echoid-s6281" xml:space="preserve">Nam NG. </s>
  <s xml:id="echoid-s6282" xml:space="preserve">NE + CE. </s>
  <s xml:id="echoid-s6283" xml:space="preserve">CG <lb/>+ AN. </s>
  <s xml:id="echoid-s6284" xml:space="preserve">AE = NG. </s>
  <s xml:id="echoid-s6285" xml:space="preserve">NE + NE. </s>
  <s xml:id="echoid-s6286" xml:space="preserve">L. </s>
  <s xml:id="echoid-s6287" xml:space="preserve">+ L. </s>
  <s xml:id="echoid-s6288" xml:space="preserve">M = NG. </s>
  <s xml:id="echoid-s6289" xml:space="preserve">M.</s>
  <s xml:id="echoid-s6290" xml:space="preserve">
<pb o="94" file="0112" n="112" rhead=""/>
unde Problematis conſtructio, ſeu puncti Z dererminatio habetur.</s>
  <s xml:id="echoid-s6291" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6292" xml:space="preserve">XXVI. </s>
  <s xml:id="echoid-s6293" xml:space="preserve">Subnectam &amp; </s>
  <s xml:id="echoid-s6294" xml:space="preserve">ab amico communicatam (aliâ methodo <lb/>
<anchor type="note" xlink:label="note-0112-01a" xlink:href="note-0112-01"/>
repertam ab ipſo, concinneque demonſtratam) conſtructionem: <lb/></s>
  <s xml:id="echoid-s6295" xml:space="preserve">Duc NR incidenti AN perpendicularem, &amp; </s>
  <s xml:id="echoid-s6296" xml:space="preserve">ſecantem axin in R. </s>
  <s xml:id="echoid-s6297" xml:space="preserve"><lb/>Fac NP. </s>
  <s xml:id="echoid-s6298" xml:space="preserve">N π :</s>
  <s xml:id="echoid-s6299" xml:space="preserve">: NR. </s>
  <s xml:id="echoid-s6300" xml:space="preserve">T. </s>
  <s xml:id="echoid-s6301" xml:space="preserve">duc N Qrefracto NK perpendicularem, <lb/>&amp; </s>
  <s xml:id="echoid-s6302" xml:space="preserve">æqualem ipſi T; </s>
  <s xml:id="echoid-s6303" xml:space="preserve">denique jungatur QC; </s>
  <s xml:id="echoid-s6304" xml:space="preserve">hæc producta ſecabit <lb/>NK in foco quæſito Z.</s>
  <s xml:id="echoid-s6305" xml:space="preserve"/>
</p>
<div xml:id="echoid-div162" type="float" level="2" n="50">
<note position="left" xlink:label="note-0112-01" xlink:href="note-0112-01a" xml:space="preserve">Fig. 147.</note>
</div>
<p>
  <s xml:id="echoid-s6306" xml:space="preserve">XXVII. </s>
  <s xml:id="echoid-s6307" xml:space="preserve">Hujuſmodi verò punctum Z eſſe locum ipſiſſimum ima-<lb/>ginis puncti A, oculo apparentis in ipſa N π conſtituto, ſæpiùs expo-<lb/>ſitæ rationes manifeſtant.</s>
  <s xml:id="echoid-s6308" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6309" xml:space="preserve">XXVIII. </s>
  <s xml:id="echoid-s6310" xml:space="preserve">Attendenti porrò conſtabit, ſiquidem fuerit <anchor type="note" xlink:href="" symbol="*"/>NG ad M
<anchor type="note" xlink:label="note-0112-02a" xlink:href="note-0112-02"/>
ratio æqualitatis, quòd punctum Z infinito à puncto G, vel N inter-<lb/>vallo diſtabit; </s>
  <s xml:id="echoid-s6311" xml:space="preserve">ſeu proximi radio N π refracti ipſi N π paralleli erunt; <lb/></s>
  <s xml:id="echoid-s6312" xml:space="preserve">ſin ratio NG ad M ſit majoris inæqualitatis, quòd punctum Z exiſtet <lb/>infra G, vel in NG antrorſum protracta; </s>
  <s xml:id="echoid-s6313" xml:space="preserve">verùm denuò ſi NG &amp;</s>
  <s xml:id="echoid-s6314" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6315" xml:space="preserve">M, <lb/>quod punctum Z ſupra N, vel in NG retrò tractâ verſatur. </s>
  <s xml:id="echoid-s6316" xml:space="preserve">Hæc <lb/>ſuffecerit innuiſſe. </s>
  <s xml:id="echoid-s6317" xml:space="preserve">Hinc etiam poſticæ circuli partis illuminatæ quan-<lb/>titas utcunque poſſit determinari. </s>
  <s xml:id="echoid-s6318" xml:space="preserve">ſed ad locum Solidum res ſpectat, <lb/>ipsámque proinde miſſam facio.</s>
  <s xml:id="echoid-s6319" xml:space="preserve"/>
</p>
<div xml:id="echoid-div163" type="float" level="2" n="51">
<note symbol="*" position="left" xlink:label="note-0112-02" xlink:href="note-0112-02a" xml:space="preserve">_In Num. 25._</note>
</div>
<p>
  <s xml:id="echoid-s6320" xml:space="preserve">XXIX. </s>
  <s xml:id="echoid-s6321" xml:space="preserve">Inſeremus autem hîc _Phænomeni_ cujuſdam ſatìs obvii, quód-<lb/>
<anchor type="note" xlink:label="note-0112-03a" xlink:href="note-0112-03"/>
que nonnullis forſan (_utpote communibus Opticæ decretis apparenter_ <lb/>_adverſum)_ mirabile videatur, explicationem. </s>
  <s xml:id="echoid-s6322" xml:space="preserve">Sit lucidi puncti A <lb/>(modicè diſtantis, &amp; </s>
  <s xml:id="echoid-s6323" xml:space="preserve">vividè radios ejaculantis) ad arcum circularem <lb/>MBN (ab axe A B biſectum) imago, ſeufocus Z; </s>
  <s xml:id="echoid-s6324" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6325" xml:space="preserve">per Z, ad ipſam <lb/>AZ perpendicularis traducta concipiatur linea XY. </s>
  <s xml:id="echoid-s6326" xml:space="preserve">porrò, deſu-<lb/>matur aliud punctum remotius E; </s>
  <s xml:id="echoid-s6327" xml:space="preserve">liquet ejus imaginem citra punctum <lb/>Z (centrum verſus) jacere; </s>
  <s xml:id="echoid-s6328" xml:space="preserve">ductis itaque rectis EM, EN, harum <lb/>refracti adhuc altiùs ſe interſecant, puta ad K; </s>
  <s xml:id="echoid-s6329" xml:space="preserve">productæque MK, <lb/>NK lineam XY ſecent punctis O, P. </s>
  <s xml:id="echoid-s6330" xml:space="preserve">quinetiam ulterius accipiatur <lb/>punctum F; </s>
  <s xml:id="echoid-s6331" xml:space="preserve">ductarúmque rectarum FM, FN refracti ſint ML, NL; <lb/></s>
  <s xml:id="echoid-s6332" xml:space="preserve">lineæ XY occurrentes ad puncta R, S; </s>
  <s xml:id="echoid-s6333" xml:space="preserve">quibus peractis manifeſtum eſt <lb/>intervallum RS ipſo OP majus eſſe. </s>
  <s xml:id="echoid-s6334" xml:space="preserve">Hinc facilis habetur ratio, cur <lb/>punctum lucidum (velut _ardens lucerna,_ vel _Imago Solis_ ad _Speculum_ <lb/>aut _lentem diaphanam_ effecta, (quin &amp; </s>
  <s xml:id="echoid-s6335" xml:space="preserve">ſtellæ fixæ) quæ propter exi-<lb/>guitatem ſuam apparentem punctorum ad inſtar haberi poſſunt) quò <lb/>a diſtinctæ viſionis loco longiùs amovetur, eò (contra quàm in aliis
<pb o="95" file="0113" n="113" rhead=""/>
viſibilibus obvenit) majus apparet. </s>
  <s xml:id="echoid-s6336" xml:space="preserve">Nam ſi arcus MNB oculi ſu-<lb/>
<anchor type="note" xlink:label="note-0113-01a" xlink:href="note-0113-01"/>
perficiem repræſentet, _(pupilli amplitudini reſpondentem)_ linea XY <lb/>fundum oculi, A locum diſtinctæ viſionis; </s>
  <s xml:id="echoid-s6337" xml:space="preserve">ejuſmodi lucens ad A po-<lb/>ſitum ſatìs anguſtum circa Z ſpatium illuſtrabit; </s>
  <s xml:id="echoid-s6338" xml:space="preserve">ad E verò conſtitu-<lb/>tum, validè radios vibrans, totum coruſcatione ſuâ ſpatium OP <lb/>afficiet; </s>
  <s xml:id="echoid-s6339" xml:space="preserve">ad F denique collocatum adhuc majus intervallum R S per-<lb/>cellet, indéque grandiorem ſui ſpeciem exhibebit. </s>
  <s xml:id="echoid-s6340" xml:space="preserve">In placidè verò <lb/>lucem remittentibus aliter ſe habet, quoniam pauciores, &amp; </s>
  <s xml:id="echoid-s6341" xml:space="preserve">langui-<lb/>diùs agentes qui extremis O, P vel R, Sallabuntur radii nullam ſui <lb/>perceptionem excitant.</s>
  <s xml:id="echoid-s6342" xml:space="preserve"/>
</p>
<div xml:id="echoid-div164" type="float" level="2" n="52">
<note position="left" xlink:label="note-0112-03" xlink:href="note-0112-03a" xml:space="preserve">Fig. 148.</note>
<note position="right" xlink:label="note-0113-01" xlink:href="note-0113-01a" xml:space="preserve">Fig. 148.</note>
</div>
<p>
  <s xml:id="echoid-s6343" xml:space="preserve">Eo lubentius hanc, adeò perſpicuam, hujuſmodi _Phænomeγων_ aſſig-<lb/>namus rationem, quoniam in eorum reddendis cauſis ità titubat magnus <lb/>ille _Galilæus,_ neſcio quos, ex refractionibus, relectionibúsve quibuſdam <lb/>commentitiis oriundos, aſcititios ſuggerens cincinnos.</s>
  <s xml:id="echoid-s6344" xml:space="preserve">‖</s>
</p>
<p>
  <s xml:id="echoid-s6345" xml:space="preserve">XXX. </s>
  <s xml:id="echoid-s6346" xml:space="preserve">Quin hic tandem _Dioptricam ſimplicens circularem claude_-<lb/>_mus,_ quam utcunque quàm pauciſſimis ità complexi ſumus, ut præ-<lb/>cipua ſaltem (quæ videbantur) &amp; </s>
  <s xml:id="echoid-s6347" xml:space="preserve">notatu digniora perſtrinxerimus. <lb/></s>
  <s xml:id="echoid-s6348" xml:space="preserve">priùs autem _Catoptricam cir cularem_; </s>
  <s xml:id="echoid-s6349" xml:space="preserve">nec non utramque, tam Di-<lb/>_optricam_ quàm _Catoptricam,_ planam, quantum inſtituto noſtro viſum <lb/>eſt congruere, pertractavimus. </s>
  <s xml:id="echoid-s6350" xml:space="preserve">quibus perfuncto mihi propoſitum <lb/>aliquando fuit ad curvas alias, conicas præſertim ſectiones, haud diſſi-<lb/>mili methodo pertentandas cogitationem extendere. </s>
  <s xml:id="echoid-s6351" xml:space="preserve">Sed enim, cùm <lb/>in his tricis _Geometricis_ etiamnum ſatìs ſupérque commoratus ſim; </s>
  <s xml:id="echoid-s6352" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s6353" xml:space="preserve">præter ea quæ circa conicas ſectiones à nobis pridem inſinuata ſunt <lb/>(quæ &amp; </s>
  <s xml:id="echoid-s6354" xml:space="preserve">ab aliis luculentè tractata proſtant) reliqua non ità magnum <lb/>uſum ſpondeant; </s>
  <s xml:id="echoid-s6355" xml:space="preserve">contentus haſce primarias, in uſu maximè poſitas, <lb/>&amp; </s>
  <s xml:id="echoid-s6356" xml:space="preserve">uſui præſertim accommodatas ſuperficies, ultra paullò quàm hacte-<lb/>nùs attentatum aut peractum ſcirem, excuſſiſſe; </s>
  <s xml:id="echoid-s6357" xml:space="preserve">cæteras omninò <lb/>miſſas faciam.</s>
  <s xml:id="echoid-s6358" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6359" xml:space="preserve">XXXI. </s>
  <s xml:id="echoid-s6360" xml:space="preserve">Porrò, quoad inflectiones iſtas, quos pluribus ſucceſſivè <lb/>planis, aut Sphæricis Superficiebus, utcunque conſtitutis aut compoſitis, <lb/>incidentes ſubeunt radii; </s>
  <s xml:id="echoid-s6361" xml:space="preserve">quæ conveniunt illis Symptomata, poſſunt ea <lb/>de præmiſſis elici; </s>
  <s xml:id="echoid-s6362" xml:space="preserve">quorum certè præcipuum eſt, quod apparentis <lb/>puncti locum reſpicit ab inflectionibus ad iſtas ſuperficies factis reſul-<lb/>tantem; </s>
  <s xml:id="echoid-s6363" xml:space="preserve">in hoc enim indagando, determinandóque potiſſimùm hæ <lb/>diſquiſitiones verſantur; </s>
  <s xml:id="echoid-s6364" xml:space="preserve">Hunc igitur ſaltem definitum exhibebimus, <lb/>idque ſatìs commodè, ex uno quodam Theoremate, ſeu regula gene-<lb/>rali; </s>
  <s xml:id="echoid-s6365" xml:space="preserve">cui exempla quædam, communis usûs in gratiam ſelecta, eorúm-<lb/>que qui in hæc inciderit minuendo labori præſertim comparata, ſubjun-<lb/>gemus. </s>
  <s xml:id="echoid-s6366" xml:space="preserve">Iſta verò, nè jam tædio Sinus, ſequenti reſervamus.</s>
  <s xml:id="echoid-s6367" xml:space="preserve"/>
</p>
<pb o="96" file="0114" n="114"/>
</div>
<div xml:id="echoid-div166" type="section" level="1" n="20">
<head xml:id="echoid-head23" xml:space="preserve"><emph style="sc">Lect.</emph> XIV.</head>
<p style="it">
  <s xml:id="echoid-s6368" xml:space="preserve">_I.</s>
  <s xml:id="echoid-s6369" xml:space="preserve">S_ub pracedentis calcem, Regulam pollicebamur, exemplis ſtipa-<lb/>tam, ex qua punctorum è variis inflectionibus reſultantes, ima-<lb/>gines dignoſcantur. </s>
  <s xml:id="echoid-s6370" xml:space="preserve">iſlam nunc exhibemus quàm ſimplicimè conceptam.</s>
  <s xml:id="echoid-s6371" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6372" xml:space="preserve">Sit ABEFO radius principalis, puncti radiantis A ſpeciem per <lb/>
<anchor type="note" xlink:label="note-0114-01a" xlink:href="note-0114-01"/>
oculi centrum O deferens, ex incidente primo AB, &amp; </s>
  <s xml:id="echoid-s6373" xml:space="preserve">inflexis BE, <lb/>EF, FO (in directum aut ſecùs diſpoſitis) conſtans; </s>
  <s xml:id="echoid-s6374" xml:space="preserve">tum puncti A <lb/>reſpectu oculi in recta B E poſiti, &amp; </s>
  <s xml:id="echoid-s6375" xml:space="preserve">ex inflectione ad ſuperficiem B <lb/>reſultans (è præmiſſis utique deſignabilis) imago ſit Z. </s>
  <s xml:id="echoid-s6376" xml:space="preserve">item hujus Z <lb/>(quod jam veluti radians concipiatur) reſpectu oculi in recta E F <lb/>conſtituti, &amp; </s>
  <s xml:id="echoid-s6377" xml:space="preserve">ab inflectione ad ſuperficiem E emergens imago ſit Y; <lb/></s>
  <s xml:id="echoid-s6378" xml:space="preserve">demùm puncti Y (tanquam in ſuperficiem F radiantis) reſpectu oculi <lb/>in FO collocati ſit imago X. </s>
  <s xml:id="echoid-s6379" xml:space="preserve">erit hoc punctum Ximago cunctis ab <lb/>his inflectionibus proveniens. </s>
  <s xml:id="echoid-s6380" xml:space="preserve">neque ſecùs quotcunque fuerint inflecti-<lb/>ones ſeſe res habebit; </s>
  <s xml:id="echoid-s6381" xml:space="preserve">enimverò ſemper ex illa tali poſtrema inflectione <lb/>reſultans imago, eadem erit cum illa, quam omnes exhibent.</s>
  <s xml:id="echoid-s6382" xml:space="preserve"/>
</p>
<div xml:id="echoid-div166" type="float" level="2" n="1">
<note position="left" xlink:label="note-0114-01" xlink:href="note-0114-01a" xml:space="preserve">Fig. 149, <lb/>150.</note>
</div>
<p>
  <s xml:id="echoid-s6383" xml:space="preserve">Hujus effati veritas è conſtructione ſatìs apparet; </s>
  <s xml:id="echoid-s6384" xml:space="preserve">è qua facilè colli-<lb/>gitur proximorum ipſi AB incidentium hinc indè radiorum inflexos <lb/>tandem circa punctum Xipſum FX interſecare. </s>
  <s xml:id="echoid-s6385" xml:space="preserve">vel ità rem collegeris: <lb/></s>
  <s xml:id="echoid-s6386" xml:space="preserve">punctum Z eſt puncti A imago; </s>
  <s xml:id="echoid-s6387" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6388" xml:space="preserve">punctum Y ipſius Z; </s>
  <s xml:id="echoid-s6389" xml:space="preserve">denuóque <lb/>punctum Xipſius Y; </s>
  <s xml:id="echoid-s6390" xml:space="preserve">itaque punctum X ipſius A imago erit, qualem <lb/>nempe res hîc fert, remota. </s>
  <s xml:id="echoid-s6391" xml:space="preserve">Strictiore longiuſculo diſcurſu poſſet hoc <lb/>comprobari, ſed quorſum rem ſatìs claram intricare?</s>
  <s xml:id="echoid-s6392" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6393" xml:space="preserve">II. </s>
  <s xml:id="echoid-s6394" xml:space="preserve">Exempla jam, quæ dixi, ſeu è præmiſſis deducta conſectaria <lb/>ſubnectam. </s>
  <s xml:id="echoid-s6395" xml:space="preserve">Notetur autem imagines, quæ in iis pròponuntur deſig-<lb/>nandæ, oculum reſpicere Centrum habentem in ipſo radiationis axe <lb/>(qualis eſt recta BD) conſtitutum. </s>
  <s xml:id="echoid-s6396" xml:space="preserve">item diverſarum ſuperficierum <lb/>ac radiationum axes ſibimet in directum poni. </s>
  <s xml:id="echoid-s6397" xml:space="preserve">præſumatur etiam in <lb/>refractionibus ex aere factis ad vitrum fore I. </s>
  <s xml:id="echoid-s6398" xml:space="preserve">R :</s>
  <s xml:id="echoid-s6399" xml:space="preserve">: 5. </s>
  <s xml:id="echoid-s6400" xml:space="preserve">3; </s>
  <s xml:id="echoid-s6401" xml:space="preserve">ad aquam <lb/>vero fore I. </s>
  <s xml:id="echoid-s6402" xml:space="preserve">R :</s>
  <s xml:id="echoid-s6403" xml:space="preserve">: 4. </s>
  <s xml:id="echoid-s6404" xml:space="preserve">3. </s>
  <s xml:id="echoid-s6405" xml:space="preserve">(hæ nempe rationes veris probè congruæ
<pb o="97" file="0115" n="115" rhead=""/>
deprehenduntur). </s>
  <s xml:id="echoid-s6406" xml:space="preserve">addo, confuſionis evitandæ causâ ſymbolum I <lb/>dehinc in his exemplis perpetuò majorem proportionis refractiones <lb/>dimetientis terminum denotare, quocunque de medio in quodcunque <lb/>peragatur refractio. </s>
  <s xml:id="echoid-s6407" xml:space="preserve">porrò, medium primum infringens perpetuò <lb/>denſius intelligatur rariori circundatum. </s>
  <s xml:id="echoid-s6408" xml:space="preserve">item, in figuris appoſitis <lb/>litera C denotat centrum anterioris circuli, K centrum poſterioris; <lb/></s>
  <s xml:id="echoid-s6409" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6410" xml:space="preserve">B verticem anterioris, D verticem poſterioris; </s>
  <s xml:id="echoid-s6411" xml:space="preserve">denuò deſignat Y <lb/>locum imaginis quæſitam.</s>
  <s xml:id="echoid-s6412" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6413" xml:space="preserve">Hiſce præmonitis, primum de longinquo radiantium, ſeu paralle-<lb/>los ejicientium radios punctorum imagines, pro lentium varietate, ſic <lb/>determinantur.</s>
  <s xml:id="echoid-s6414" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6415" xml:space="preserve">_I._ </s>
  <s xml:id="echoid-s6416" xml:space="preserve">Ad lentem plano-convexam.</s>
  <s xml:id="echoid-s6417" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 151.</note>
<p style="it">
  <s xml:id="echoid-s6418" xml:space="preserve">_II._ </s>
  <s xml:id="echoid-s6419" xml:space="preserve">Ad lentem plano-concavam.</s>
  <s xml:id="echoid-s6420" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6421" xml:space="preserve">Fiat I- R. </s>
  <s xml:id="echoid-s6422" xml:space="preserve">R :</s>
  <s xml:id="echoid-s6423" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6424" xml:space="preserve">DY.</s>
  <s xml:id="echoid-s6425" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6426" xml:space="preserve">_In Vitro_ eſt DY = {3/2} KD.</s>
  <s xml:id="echoid-s6427" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6428" xml:space="preserve">_In Aqua_ eſt DY = 3 KD.</s>
  <s xml:id="echoid-s6429" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6430" xml:space="preserve">_III._ </s>
  <s xml:id="echoid-s6431" xml:space="preserve">Ad lentem convexo-planam.</s>
  <s xml:id="echoid-s6432" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 151, <lb/>152.</note>
<p style="it">
  <s xml:id="echoid-s6433" xml:space="preserve">_IV._ </s>
  <s xml:id="echoid-s6434" xml:space="preserve">Ad lentem concavo-planam.</s>
  <s xml:id="echoid-s6435" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6436" xml:space="preserve">Fiat {I - R. </s>
  <s xml:id="echoid-s6437" xml:space="preserve">I :</s>
  <s xml:id="echoid-s6438" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6439" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s6440" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6441" xml:space="preserve">\\ I. </s>
  <s xml:id="echoid-s6442" xml:space="preserve">R :</s>
  <s xml:id="echoid-s6443" xml:space="preserve">: DZ. </s>
  <s xml:id="echoid-s6444" xml:space="preserve">DY.</s>
  <s xml:id="echoid-s6445" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6446" xml:space="preserve">_In Vitro_ DY = {3/2} BC - {3/5} BD.</s>
  <s xml:id="echoid-s6447" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6448" xml:space="preserve">_In Aqua_ DY = 3 BC - {3/4} BD.</s>
  <s xml:id="echoid-s6449" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6450" xml:space="preserve">_V._ </s>
  <s xml:id="echoid-s6451" xml:space="preserve">Ad lentem convexo convexam.</s>
  <s xml:id="echoid-s6452" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 152.</note>
<p style="it">
  <s xml:id="echoid-s6453" xml:space="preserve">_VI._ </s>
  <s xml:id="echoid-s6454" xml:space="preserve">Ad lentem concauo-concauam.</s>
  <s xml:id="echoid-s6455" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6456" xml:space="preserve">Fiat {I- R. </s>
  <s xml:id="echoid-s6457" xml:space="preserve">I :</s>
  <s xml:id="echoid-s6458" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6459" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s6460" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6461" xml:space="preserve">\\ {I/R} KZ - DZ. </s>
  <s xml:id="echoid-s6462" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6463" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6464" xml:space="preserve">DY.</s>
  <s xml:id="echoid-s6465" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6466" xml:space="preserve">_Coroll._ </s>
  <s xml:id="echoid-s6467" xml:space="preserve">Adintegram Spharam.</s>
  <s xml:id="echoid-s6468" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6469" xml:space="preserve">Fiat 2 I - 2 R. </s>
  <s xml:id="echoid-s6470" xml:space="preserve">I :</s>
  <s xml:id="echoid-s6471" xml:space="preserve">: CD. </s>
  <s xml:id="echoid-s6472" xml:space="preserve">CY.</s>
  <s xml:id="echoid-s6473" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6474" xml:space="preserve">_VII._ </s>
  <s xml:id="echoid-s6475" xml:space="preserve">Adlentem conuexo-concauam.</s>
  <s xml:id="echoid-s6476" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 152, <lb/>153.</note>
<p style="it">
  <s xml:id="echoid-s6477" xml:space="preserve">_VIII._ </s>
  <s xml:id="echoid-s6478" xml:space="preserve">Ad lentem concauo conuexam.</s>
  <s xml:id="echoid-s6479" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6480" xml:space="preserve">Fiat I - R. </s>
  <s xml:id="echoid-s6481" xml:space="preserve">I :</s>
  <s xml:id="echoid-s6482" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6483" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s6484" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s6485" xml:space="preserve"/>
</p>
<pb o="98" file="0116" n="116" rhead=""/>
<p>
  <s xml:id="echoid-s6486" xml:space="preserve">1. </s>
  <s xml:id="echoid-s6487" xml:space="preserve">Si punctum Z cadat inter C, &amp; </s>
  <s xml:id="echoid-s6488" xml:space="preserve">K, fac DZ + {I/R} KZ. </s>
  <s xml:id="echoid-s6489" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6490" xml:space="preserve">: <lb/>
<anchor type="note" xlink:label="note-0116-01a" xlink:href="note-0116-01"/>
DK. </s>
  <s xml:id="echoid-s6491" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6492" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6493" xml:space="preserve">cape DY ad partes lentis verſus K.</s>
  <s xml:id="echoid-s6494" xml:space="preserve"/>
</p>
<div xml:id="echoid-div167" type="float" level="2" n="2">
<note position="left" xlink:label="note-0116-01" xlink:href="note-0116-01a" xml:space="preserve">Fig. 152, <lb/>153.</note>
</div>
<p>
  <s xml:id="echoid-s6495" xml:space="preserve">2. </s>
  <s xml:id="echoid-s6496" xml:space="preserve">Si punctum Z cadat extra CK, &amp; </s>
  <s xml:id="echoid-s6497" xml:space="preserve">ſit inſuper DZ &amp;</s>
  <s xml:id="echoid-s6498" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6499" xml:space="preserve">{I/R} KZ, <lb/>fac DZ &amp;</s>
  <s xml:id="echoid-s6500" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6501" xml:space="preserve">{I/R} KZ. </s>
  <s xml:id="echoid-s6502" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6503" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6504" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6505" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6506" xml:space="preserve">cape DY ad partes lentis <lb/>verſus K.</s>
  <s xml:id="echoid-s6507" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6508" xml:space="preserve">3. </s>
  <s xml:id="echoid-s6509" xml:space="preserve">Si DZ = {I/R} KZ, imago Y infinitè diſtabit.</s>
  <s xml:id="echoid-s6510" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6511" xml:space="preserve">4. </s>
  <s xml:id="echoid-s6512" xml:space="preserve">Si DZ &amp;</s>
  <s xml:id="echoid-s6513" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6514" xml:space="preserve">{I/R} KZ; </s>
  <s xml:id="echoid-s6515" xml:space="preserve">fiat {I/R} KZ - DZ. </s>
  <s xml:id="echoid-s6516" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6517" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6518" xml:space="preserve">DY, &amp; </s>
  <s xml:id="echoid-s6519" xml:space="preserve"><lb/>cape DY ad partes lentis adverſas ipſi K.</s>
  <s xml:id="echoid-s6520" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6521" xml:space="preserve">De ſenſibiliter autem propinqua diſtantia radiantium ſeu divergentes <lb/>radios emittentium punctorum (qualia ſemper deſignat punctum A) <lb/>imagines (ut &amp; </s>
  <s xml:id="echoid-s6522" xml:space="preserve">illæ quas ad ejuſmodi puncta convergentes efficiunt <lb/>radii) hoc pacto determinantur.</s>
  <s xml:id="echoid-s6523" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6524" xml:space="preserve">_I._ </s>
  <s xml:id="echoid-s6525" xml:space="preserve">Ad lentem plano-planam diverg.</s>
  <s xml:id="echoid-s6526" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Fig. 154, <lb/>155.</note>
<p style="it">
  <s xml:id="echoid-s6527" xml:space="preserve">_II._ </s>
  <s xml:id="echoid-s6528" xml:space="preserve">Ad lentem plano-planam converg.</s>
  <s xml:id="echoid-s6529" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6530" xml:space="preserve">Fiat {R. </s>
  <s xml:id="echoid-s6531" xml:space="preserve">I :</s>
  <s xml:id="echoid-s6532" xml:space="preserve">: AB. </s>
  <s xml:id="echoid-s6533" xml:space="preserve">BZ, &amp; </s>
  <s xml:id="echoid-s6534" xml:space="preserve">\\ I. </s>
  <s xml:id="echoid-s6535" xml:space="preserve">R :</s>
  <s xml:id="echoid-s6536" xml:space="preserve">: DZ. </s>
  <s xml:id="echoid-s6537" xml:space="preserve">DY.</s>
  <s xml:id="echoid-s6538" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6539" xml:space="preserve">Breviùs. </s>
  <s xml:id="echoid-s6540" xml:space="preserve">Fiat I. </s>
  <s xml:id="echoid-s6541" xml:space="preserve">I - R :</s>
  <s xml:id="echoid-s6542" xml:space="preserve">: BD. </s>
  <s xml:id="echoid-s6543" xml:space="preserve">AY.</s>
  <s xml:id="echoid-s6544" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6545" xml:space="preserve">_III._ </s>
  <s xml:id="echoid-s6546" xml:space="preserve">Ad lentem plano-convexam diverg.</s>
  <s xml:id="echoid-s6547" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Fig. 156.</note>
<p style="it">
  <s xml:id="echoid-s6548" xml:space="preserve">_IV._ </s>
  <s xml:id="echoid-s6549" xml:space="preserve">Ad lentem plano concavam converg.</s>
  <s xml:id="echoid-s6550" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6551" xml:space="preserve">Fiat R. </s>
  <s xml:id="echoid-s6552" xml:space="preserve">I :</s>
  <s xml:id="echoid-s6553" xml:space="preserve">: AB. </s>
  <s xml:id="echoid-s6554" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s6555" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6556" xml:space="preserve">cum Z cadit</s>
</p>
<p>
  <s xml:id="echoid-s6557" xml:space="preserve">1. </s>
  <s xml:id="echoid-s6558" xml:space="preserve">Extra DK, ſi {I/R} KZ &amp;</s>
  <s xml:id="echoid-s6559" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6560" xml:space="preserve">DZ; </s>
  <s xml:id="echoid-s6561" xml:space="preserve">fac {I/R} KZ - DZ. </s>
  <s xml:id="echoid-s6562" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6563" xml:space="preserve">: <lb/>DK. </s>
  <s xml:id="echoid-s6564" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6565" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6566" xml:space="preserve">cape DY ad partes lentis adverſus A.</s>
  <s xml:id="echoid-s6567" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6568" xml:space="preserve">2. </s>
  <s xml:id="echoid-s6569" xml:space="preserve">Si {I/R} KZ = DZ; </s>
  <s xml:id="echoid-s6570" xml:space="preserve">imago diſtabit infinitè.</s>
  <s xml:id="echoid-s6571" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6572" xml:space="preserve">3. </s>
  <s xml:id="echoid-s6573" xml:space="preserve">Si {I/R} KZ &amp;</s>
  <s xml:id="echoid-s6574" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6575" xml:space="preserve">DZ; </s>
  <s xml:id="echoid-s6576" xml:space="preserve">fac DZ - {I/R} KZ. </s>
  <s xml:id="echoid-s6577" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6578" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6579" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6580" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6581" xml:space="preserve"><lb/>eape DY verſus A.</s>
  <s xml:id="echoid-s6582" xml:space="preserve"/>
</p>
<pb o="99" file="0117" n="117" rhead=""/>
<p>
  <s xml:id="echoid-s6583" xml:space="preserve">4. </s>
  <s xml:id="echoid-s6584" xml:space="preserve">Cùm Z cadit inter puncta D, K; </s>
  <s xml:id="echoid-s6585" xml:space="preserve">fac DZ + {I/R} KZ. </s>
  <s xml:id="echoid-s6586" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6587" xml:space="preserve">: <lb/>DK. </s>
  <s xml:id="echoid-s6588" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6589" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6590" xml:space="preserve">cape DY verſus A.</s>
  <s xml:id="echoid-s6591" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6592" xml:space="preserve">_V._ </s>
  <s xml:id="echoid-s6593" xml:space="preserve">Ad lentem plano-concavam diverg.</s>
  <s xml:id="echoid-s6594" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 156, <lb/>157.</note>
<p style="it">
  <s xml:id="echoid-s6595" xml:space="preserve">_VI._ </s>
  <s xml:id="echoid-s6596" xml:space="preserve">Ad lentem plano convexam converg.</s>
  <s xml:id="echoid-s6597" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6598" xml:space="preserve">Fiat {R. </s>
  <s xml:id="echoid-s6599" xml:space="preserve">I :</s>
  <s xml:id="echoid-s6600" xml:space="preserve">: AB. </s>
  <s xml:id="echoid-s6601" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s6602" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6603" xml:space="preserve">\\ {I/R} KZ - DZ. </s>
  <s xml:id="echoid-s6604" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6605" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6606" xml:space="preserve">DY.</s>
  <s xml:id="echoid-s6607" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6608" xml:space="preserve">_VII._ </s>
  <s xml:id="echoid-s6609" xml:space="preserve">Adlentem convexo-planam diverg.</s>
  <s xml:id="echoid-s6610" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 157.</note>
<p style="it">
  <s xml:id="echoid-s6611" xml:space="preserve">_VIII._ </s>
  <s xml:id="echoid-s6612" xml:space="preserve">Ad lentem concavo-planam converg.</s>
  <s xml:id="echoid-s6613" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6614" xml:space="preserve">1. </s>
  <s xml:id="echoid-s6615" xml:space="preserve">Si AB &amp;</s>
  <s xml:id="echoid-s6616" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6617" xml:space="preserve">{R/I} AC, puncta Z, &amp; </s>
  <s xml:id="echoid-s6618" xml:space="preserve">Y ad lentis partes puncto A <lb/>adverſas reperientur, facto AB - {R/I} AC. </s>
  <s xml:id="echoid-s6619" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s6620" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6621" xml:space="preserve">BZ. </s>
  <s xml:id="echoid-s6622" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6623" xml:space="preserve"><lb/>I. </s>
  <s xml:id="echoid-s6624" xml:space="preserve">R :</s>
  <s xml:id="echoid-s6625" xml:space="preserve">: DZ. </s>
  <s xml:id="echoid-s6626" xml:space="preserve">DY.</s>
  <s xml:id="echoid-s6627" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6628" xml:space="preserve">2. </s>
  <s xml:id="echoid-s6629" xml:space="preserve">Si AB = {R/I} AC, imago infinitè diſtabit.</s>
  <s xml:id="echoid-s6630" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6631" xml:space="preserve">3. </s>
  <s xml:id="echoid-s6632" xml:space="preserve">Si AB &amp;</s>
  <s xml:id="echoid-s6633" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6634" xml:space="preserve">{R/I} AC; </s>
  <s xml:id="echoid-s6635" xml:space="preserve">deprehendentur Z, &amp; </s>
  <s xml:id="echoid-s6636" xml:space="preserve">Y verſus A, facto <lb/>{R/I} AC - AB. </s>
  <s xml:id="echoid-s6637" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s6638" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6639" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s6640" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6641" xml:space="preserve">I. </s>
  <s xml:id="echoid-s6642" xml:space="preserve">R :</s>
  <s xml:id="echoid-s6643" xml:space="preserve">: DZ. </s>
  <s xml:id="echoid-s6644" xml:space="preserve">DY.</s>
  <s xml:id="echoid-s6645" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6646" xml:space="preserve">_IX._ </s>
  <s xml:id="echoid-s6647" xml:space="preserve">Ad lentem concavo-planam diverg.</s>
  <s xml:id="echoid-s6648" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 158.</note>
<p style="it">
  <s xml:id="echoid-s6649" xml:space="preserve">_X._ </s>
  <s xml:id="echoid-s6650" xml:space="preserve">Ad lentem convexo-planam converg.</s>
  <s xml:id="echoid-s6651" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6652" xml:space="preserve">Si A cadat extra BC, fac AB - {R/I} AC. </s>
  <s xml:id="echoid-s6653" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s6654" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6655" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s6656" xml:space="preserve">ſin <lb/>A cadat inter B, &amp; </s>
  <s xml:id="echoid-s6657" xml:space="preserve">C, fac AB + {R/I} AC. </s>
  <s xml:id="echoid-s6658" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s6659" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6660" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s6661" xml:space="preserve">tum <lb/>fiat I. </s>
  <s xml:id="echoid-s6662" xml:space="preserve">R :</s>
  <s xml:id="echoid-s6663" xml:space="preserve">: DZ. </s>
  <s xml:id="echoid-s6664" xml:space="preserve">DY.</s>
  <s xml:id="echoid-s6665" xml:space="preserve"/>
</p>
<pb o="100" file="0118" n="118" rhead=""/>
<p style="it">
  <s xml:id="echoid-s6666" xml:space="preserve">_XI._ </s>
  <s xml:id="echoid-s6667" xml:space="preserve">Ad lentem convexo-convexam diverg.</s>
  <s xml:id="echoid-s6668" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Fig. 158, <lb/>159.</note>
<p style="it">
  <s xml:id="echoid-s6669" xml:space="preserve">_XII._ </s>
  <s xml:id="echoid-s6670" xml:space="preserve">Ad lentem concavo-concavam converg.</s>
  <s xml:id="echoid-s6671" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6672" xml:space="preserve">1. </s>
  <s xml:id="echoid-s6673" xml:space="preserve">Si AB &amp;</s>
  <s xml:id="echoid-s6674" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6675" xml:space="preserve">{R/I} AC, facto AB - {R/I} AC. </s>
  <s xml:id="echoid-s6676" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s6677" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6678" xml:space="preserve">BZ; <lb/></s>
  <s xml:id="echoid-s6679" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6680" xml:space="preserve">{I/R} KZ - DZ :</s>
  <s xml:id="echoid-s6681" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6682" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6683" xml:space="preserve">puncta Z, Y adverſus Acadunt.</s>
  <s xml:id="echoid-s6684" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6685" xml:space="preserve">2. </s>
  <s xml:id="echoid-s6686" xml:space="preserve">Si AB = {R/I} AC, fac I - R. </s>
  <s xml:id="echoid-s6687" xml:space="preserve">R :</s>
  <s xml:id="echoid-s6688" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6689" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6690" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6691" xml:space="preserve">cape DY <lb/>adverſus A.</s>
  <s xml:id="echoid-s6692" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6693" xml:space="preserve">3. </s>
  <s xml:id="echoid-s6694" xml:space="preserve">Si AB &amp;</s>
  <s xml:id="echoid-s6695" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6696" xml:space="preserve">{R/I} AC; </s>
  <s xml:id="echoid-s6697" xml:space="preserve">fac {R/I} AC - AB. </s>
  <s xml:id="echoid-s6698" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s6699" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6700" xml:space="preserve">BZ; <lb/></s>
  <s xml:id="echoid-s6701" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6702" xml:space="preserve">ſume BZ verſus A. </s>
  <s xml:id="echoid-s6703" xml:space="preserve">Jam cùm Z cadit extra DK, ſi primò ſit <lb/>{I/R} KZ &amp;</s>
  <s xml:id="echoid-s6704" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6705" xml:space="preserve">DZ, fac {I/R} KZ - DZ. </s>
  <s xml:id="echoid-s6706" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6707" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6708" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6709" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6710" xml:space="preserve">ſume <lb/>DY adverſus A</s>
</p>
<p>
  <s xml:id="echoid-s6711" xml:space="preserve">4. </s>
  <s xml:id="echoid-s6712" xml:space="preserve">Secundò, ſi {I/R} KZ = DZ, imago diſtabit infini è.</s>
  <s xml:id="echoid-s6713" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6714" xml:space="preserve">5. </s>
  <s xml:id="echoid-s6715" xml:space="preserve">Tertiò, ſi {I/R} KZ &amp;</s>
  <s xml:id="echoid-s6716" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6717" xml:space="preserve">DZ, fac DZ - {I/R} KZ. </s>
  <s xml:id="echoid-s6718" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6719" xml:space="preserve">: DK. <lb/></s>
  <s xml:id="echoid-s6720" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6721" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6722" xml:space="preserve">ſume DY verſus A.</s>
  <s xml:id="echoid-s6723" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6724" xml:space="preserve">6. </s>
  <s xml:id="echoid-s6725" xml:space="preserve">Quum denuò cadit Z inter D, &amp; </s>
  <s xml:id="echoid-s6726" xml:space="preserve">K, fiat DZ + {I/R} KZ. </s>
  <s xml:id="echoid-s6727" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6728" xml:space="preserve">: <lb/>DK. </s>
  <s xml:id="echoid-s6729" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6730" xml:space="preserve">ſumatúrque DY verſus A.</s>
  <s xml:id="echoid-s6731" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6732" xml:space="preserve">Corol. </s>
  <s xml:id="echoid-s6733" xml:space="preserve">Ad in regram Sphæram diυerg.</s>
  <s xml:id="echoid-s6734" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6735" xml:space="preserve">1. </s>
  <s xml:id="echoid-s6736" xml:space="preserve">Si AB + AC &amp;</s>
  <s xml:id="echoid-s6737" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6738" xml:space="preserve">{2R/I} AC; </s>
  <s xml:id="echoid-s6739" xml:space="preserve">fiat AB + AC - {2R/I} AC. <lb/></s>
  <s xml:id="echoid-s6740" xml:space="preserve">AC :</s>
  <s xml:id="echoid-s6741" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6742" xml:space="preserve">CY; </s>
  <s xml:id="echoid-s6743" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6744" xml:space="preserve">cape CY adverſus A.</s>
  <s xml:id="echoid-s6745" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6746" xml:space="preserve">2. </s>
  <s xml:id="echoid-s6747" xml:space="preserve">Si AB + AC = {2R/I} AC; </s>
  <s xml:id="echoid-s6748" xml:space="preserve">imago in infinitum abit.</s>
  <s xml:id="echoid-s6749" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6750" xml:space="preserve">3. </s>
  <s xml:id="echoid-s6751" xml:space="preserve">Si AB + AC &amp;</s>
  <s xml:id="echoid-s6752" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6753" xml:space="preserve">{2R/I} AC; </s>
  <s xml:id="echoid-s6754" xml:space="preserve">fiat {2R/I} AC - AC - AB. <lb/></s>
  <s xml:id="echoid-s6755" xml:space="preserve">AC :</s>
  <s xml:id="echoid-s6756" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6757" xml:space="preserve">CY; </s>
  <s xml:id="echoid-s6758" xml:space="preserve">capiatúrque CY verſus A.</s>
  <s xml:id="echoid-s6759" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6760" xml:space="preserve">_XIII._ </s>
  <s xml:id="echoid-s6761" xml:space="preserve">Ad lentem concavc-concavam diverg.</s>
  <s xml:id="echoid-s6762" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Fig. 159.</note>
<p style="it">
  <s xml:id="echoid-s6763" xml:space="preserve">_XIV._ </s>
  <s xml:id="echoid-s6764" xml:space="preserve">Ad lentem convexo-convexam converg.</s>
  <s xml:id="echoid-s6765" xml:space="preserve"/>
</p>
<pb o="101" file="0119" n="119" rhead=""/>
<p>
  <s xml:id="echoid-s6766" xml:space="preserve">Si A cadat extra BC, fiat AB - {R/I} AC. </s>
  <s xml:id="echoid-s6767" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s6768" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6769" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s6770" xml:space="preserve">ſin A <lb/>
<anchor type="note" xlink:label="note-0119-01a" xlink:href="note-0119-01"/>
cadat inter B, C; </s>
  <s xml:id="echoid-s6771" xml:space="preserve">fiat AB + {R/I} AC. </s>
  <s xml:id="echoid-s6772" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s6773" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6774" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s6775" xml:space="preserve">deinde fac <lb/>{I/R} KZ - DZ. </s>
  <s xml:id="echoid-s6776" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6777" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6778" xml:space="preserve">DY.</s>
  <s xml:id="echoid-s6779" xml:space="preserve"/>
</p>
<div xml:id="echoid-div168" type="float" level="2" n="3">
<note position="right" xlink:label="note-0119-01" xlink:href="note-0119-01a" xml:space="preserve">Fig. 159.</note>
</div>
<p style="it">
  <s xml:id="echoid-s6780" xml:space="preserve">Coroll- Adintegram Sphæram converg.</s>
  <s xml:id="echoid-s6781" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6782" xml:space="preserve">Si punctum A extra BC ponatur, fiat AB + {I-2R/I} AC :</s>
  <s xml:id="echoid-s6783" xml:space="preserve">: BC. <lb/></s>
  <s xml:id="echoid-s6784" xml:space="preserve">CY. </s>
  <s xml:id="echoid-s6785" xml:space="preserve">ſin A cadat inter B, &amp; </s>
  <s xml:id="echoid-s6786" xml:space="preserve">C; </s>
  <s xml:id="echoid-s6787" xml:space="preserve">fiat AB+{2R-I/I} AC. </s>
  <s xml:id="echoid-s6788" xml:space="preserve">AC :</s>
  <s xml:id="echoid-s6789" xml:space="preserve">: <lb/>BC. </s>
  <s xml:id="echoid-s6790" xml:space="preserve">CY; </s>
  <s xml:id="echoid-s6791" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6792" xml:space="preserve">cape CY ad partes centri verſus A.</s>
  <s xml:id="echoid-s6793" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6794" xml:space="preserve">_XV._ </s>
  <s xml:id="echoid-s6795" xml:space="preserve">Ad lentera convexo-concavam diverg.</s>
  <s xml:id="echoid-s6796" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 160, <lb/>161.</note>
<p style="it">
  <s xml:id="echoid-s6797" xml:space="preserve">_XVI._ </s>
  <s xml:id="echoid-s6798" xml:space="preserve">Ad lentem concavo convexam converg.</s>
  <s xml:id="echoid-s6799" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6800" xml:space="preserve">1. </s>
  <s xml:id="echoid-s6801" xml:space="preserve">Si AB &amp;</s>
  <s xml:id="echoid-s6802" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6803" xml:space="preserve">{R/I} AC; </s>
  <s xml:id="echoid-s6804" xml:space="preserve">puncta Z, &amp; </s>
  <s xml:id="echoid-s6805" xml:space="preserve">Y verſus A cadunt, facto <lb/>{R/I} AC - AB. </s>
  <s xml:id="echoid-s6806" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s6807" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6808" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s6809" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6810" xml:space="preserve">{I/R} KZ - DZ. </s>
  <s xml:id="echoid-s6811" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6812" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6813" xml:space="preserve">DY.</s>
  <s xml:id="echoid-s6814" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6815" xml:space="preserve">2.</s>
  <s xml:id="echoid-s6816" xml:space="preserve">Si AB = {R/I} AC; </s>
  <s xml:id="echoid-s6817" xml:space="preserve">fac I - R. </s>
  <s xml:id="echoid-s6818" xml:space="preserve">R :</s>
  <s xml:id="echoid-s6819" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6820" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6821" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6822" xml:space="preserve">cape DY <lb/>verſus A.</s>
  <s xml:id="echoid-s6823" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6824" xml:space="preserve">3. </s>
  <s xml:id="echoid-s6825" xml:space="preserve">Si AB &amp;</s>
  <s xml:id="echoid-s6826" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6827" xml:space="preserve">{R/I} AC; </s>
  <s xml:id="echoid-s6828" xml:space="preserve">fac AB - {R/I} AC. </s>
  <s xml:id="echoid-s6829" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s6830" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6831" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s6832" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6833" xml:space="preserve"><lb/>cape BZ adverſus A. </s>
  <s xml:id="echoid-s6834" xml:space="preserve">Jam quum Z cadit extra DK, tum primò ſi <lb/>{I/R} KZ &amp;</s>
  <s xml:id="echoid-s6835" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6836" xml:space="preserve">DZ, fac {I/R} KZ - DZ. </s>
  <s xml:id="echoid-s6837" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6838" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6839" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6840" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6841" xml:space="preserve">cape DY <lb/>verſus A.</s>
  <s xml:id="echoid-s6842" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6843" xml:space="preserve">4.</s>
  <s xml:id="echoid-s6844" xml:space="preserve">Secundò, ſi {I/R} KZ = DZ, imago infinitè diſtabit.</s>
  <s xml:id="echoid-s6845" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6846" xml:space="preserve">5. </s>
  <s xml:id="echoid-s6847" xml:space="preserve">Tertiò, ſi {I/R} KZ &amp;</s>
  <s xml:id="echoid-s6848" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6849" xml:space="preserve">DZ, fac DZ - {I/R} KZ. </s>
  <s xml:id="echoid-s6850" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6851" xml:space="preserve">: DK. <lb/></s>
  <s xml:id="echoid-s6852" xml:space="preserve">DY, &amp; </s>
  <s xml:id="echoid-s6853" xml:space="preserve">ſume DY adverſus A.</s>
  <s xml:id="echoid-s6854" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6855" xml:space="preserve">6. </s>
  <s xml:id="echoid-s6856" xml:space="preserve">Sed quando Z inter D, &amp; </s>
  <s xml:id="echoid-s6857" xml:space="preserve">K cadit; </s>
  <s xml:id="echoid-s6858" xml:space="preserve">fiat DZ + {I/R} KZ. </s>
  <s xml:id="echoid-s6859" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6860" xml:space="preserve">: <lb/>DK. </s>
  <s xml:id="echoid-s6861" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6862" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6863" xml:space="preserve">ſumatur DY adverſus A.</s>
  <s xml:id="echoid-s6864" xml:space="preserve"/>
</p>
<pb o="102" file="0120" n="120" rhead=""/>
<p style="it">
  <s xml:id="echoid-s6865" xml:space="preserve">_XVII._ </s>
  <s xml:id="echoid-s6866" xml:space="preserve">Ad lentem concavo convexam diverg.</s>
  <s xml:id="echoid-s6867" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Fig. 160, <lb/>161, 162.</note>
<p style="it">
  <s xml:id="echoid-s6868" xml:space="preserve">_XVIII._ </s>
  <s xml:id="echoid-s6869" xml:space="preserve">Ad lentem convexo-concavam converg.</s>
  <s xml:id="echoid-s6870" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6871" xml:space="preserve">Si Acadat extra BC, fiat AB - {R/I} AC. </s>
  <s xml:id="echoid-s6872" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s6873" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6874" xml:space="preserve">BZ; </s>
  <s xml:id="echoid-s6875" xml:space="preserve">ſin A <lb/>cadat inter B, &amp; </s>
  <s xml:id="echoid-s6876" xml:space="preserve">C; </s>
  <s xml:id="echoid-s6877" xml:space="preserve">fiat AB + {R/I} AC. </s>
  <s xml:id="echoid-s6878" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s6879" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s6880" xml:space="preserve">BZ.</s>
  <s xml:id="echoid-s6881" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6882" xml:space="preserve">1. </s>
  <s xml:id="echoid-s6883" xml:space="preserve">Jam cùm Z cadit extra DK, tum primò ſi {I/R} KZ &amp;</s>
  <s xml:id="echoid-s6884" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s6885" xml:space="preserve">DZ, fac <lb/>{I/R} KZ - DZ. </s>
  <s xml:id="echoid-s6886" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6887" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s6888" xml:space="preserve">DY, &amp; </s>
  <s xml:id="echoid-s6889" xml:space="preserve">cape DY adverſus A.</s>
  <s xml:id="echoid-s6890" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6891" xml:space="preserve">2. </s>
  <s xml:id="echoid-s6892" xml:space="preserve">Secundò, ſi {I/R} KZ = DZ; </s>
  <s xml:id="echoid-s6893" xml:space="preserve">imagò infinitè elongabitur.</s>
  <s xml:id="echoid-s6894" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6895" xml:space="preserve">3. </s>
  <s xml:id="echoid-s6896" xml:space="preserve">Tertiò, ſi {I/R} KZ &amp;</s>
  <s xml:id="echoid-s6897" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s6898" xml:space="preserve">DZ, fac DZ - {I/R} KZ. </s>
  <s xml:id="echoid-s6899" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6900" xml:space="preserve">: DK. <lb/></s>
  <s xml:id="echoid-s6901" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6902" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6903" xml:space="preserve">ſume DY verſus A.</s>
  <s xml:id="echoid-s6904" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6905" xml:space="preserve">4. </s>
  <s xml:id="echoid-s6906" xml:space="preserve">Sed quando Z inter D, &amp; </s>
  <s xml:id="echoid-s6907" xml:space="preserve">K cadit, fiat DZ + {I/R} KZ. </s>
  <s xml:id="echoid-s6908" xml:space="preserve">DZ :</s>
  <s xml:id="echoid-s6909" xml:space="preserve">: <lb/>DK. </s>
  <s xml:id="echoid-s6910" xml:space="preserve">DY; </s>
  <s xml:id="echoid-s6911" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6912" xml:space="preserve">accipiatur DY verſus A.</s>
  <s xml:id="echoid-s6913" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6914" xml:space="preserve">Hiſce ſubnectam ſequentia; </s>
  <s xml:id="echoid-s6915" xml:space="preserve">non contemnendum in _engyſcopicis_ <lb/>uſum præ ſe ferentia _Problemata._</s>
  <s xml:id="echoid-s6916" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6917" xml:space="preserve">I. </s>
  <s xml:id="echoid-s6918" xml:space="preserve">_Dati puncti propinqui A perfectam imaginem per lentem concavo-_ <lb/>
<anchor type="note" xlink:label="note-0120-02a" xlink:href="note-0120-02"/>
_convexam in aliud datum punctum Z lenti vicinius projicere._ </s>
  <s xml:id="echoid-s6919" xml:space="preserve">(per-<lb/>fectam imaginem intelligo, quæ reſultat ex omnibus, quos ipſum A <lb/>diffundit, radiis in ipſa readunatis.)</s>
  <s xml:id="echoid-s6920" xml:space="preserve"/>
</p>
<div xml:id="echoid-div169" type="float" level="2" n="4">
<note position="left" xlink:label="note-0120-02" xlink:href="note-0120-02a" xml:space="preserve">Fig. 163.</note>
</div>
<p>
  <s xml:id="echoid-s6921" xml:space="preserve">Fiat I - R. </s>
  <s xml:id="echoid-s6922" xml:space="preserve">R :</s>
  <s xml:id="echoid-s6923" xml:space="preserve">: AZ. </s>
  <s xml:id="echoid-s6924" xml:space="preserve">ZB. </s>
  <s xml:id="echoid-s6925" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6926" xml:space="preserve">dividatur ZB in C, ut ſit CB. <lb/></s>
  <s xml:id="echoid-s6927" xml:space="preserve">CZ :</s>
  <s xml:id="echoid-s6928" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s6929" xml:space="preserve">R. </s>
  <s xml:id="echoid-s6930" xml:space="preserve">tum centro C deſcribatur circulus EBF. </s>
  <s xml:id="echoid-s6931" xml:space="preserve">item centro Z <lb/>intervallo quovis ZD (majoriquam ZB) deſcribatur circulus GDH; </s>
  <s xml:id="echoid-s6932" xml:space="preserve"><lb/>factum erit; </s>
  <s xml:id="echoid-s6933" xml:space="preserve">nempe lens EFGH puncti A perfectam imaginem in <lb/>punctum Z projiciet.</s>
  <s xml:id="echoid-s6934" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6935" xml:space="preserve">Nota, datâ CB puncta A, Z è propoſitis facilè determinari.</s>
  <s xml:id="echoid-s6936" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6937" xml:space="preserve">In vitro, ſi CB = 15, erit {ZC = 9 \\ ZB = 24} &amp; </s>
  <s xml:id="echoid-s6938" xml:space="preserve">{AZ = 16. </s>
  <s xml:id="echoid-s6939" xml:space="preserve">\\ AB = 40. <lb/></s>
  <s xml:id="echoid-s6940" xml:space="preserve">Adnotetur etiam per lentem EGHF ad Z tendentes radios ad A <lb/>refringi.</s>
  <s xml:id="echoid-s6941" xml:space="preserve"/>
</p>
<pb o="103" file="0121" n="121" rhead=""/>
<p>
  <s xml:id="echoid-s6942" xml:space="preserve">_Hujuſmodi Vitrum Myopes juvat;_ </s>
  <s xml:id="echoid-s6943" xml:space="preserve">pro quibus ità conſtruatur: <lb/></s>
  <s xml:id="echoid-s6944" xml:space="preserve">ſit ZD diſtantia, ad quam optimè cernunt; </s>
  <s xml:id="echoid-s6945" xml:space="preserve">ſumatúrque ZB utcun-<lb/>que paullo minor quàm ZD; </s>
  <s xml:id="echoid-s6946" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6947" xml:space="preserve">fiat CB = {5/8} ZB; </s>
  <s xml:id="echoid-s6948" xml:space="preserve">tum centro C <lb/>per B deſcribatur circulus EBF, &amp; </s>
  <s xml:id="echoid-s6949" xml:space="preserve">centro Z per D circulus GDH <lb/>deſcribatur; </s>
  <s xml:id="echoid-s6950" xml:space="preserve">ipſi (Superficiei GDH oculum admoventes) punctum <lb/>A diſtinctè ſpectabunt, velut ad Z ſitum.</s>
  <s xml:id="echoid-s6951" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6952" xml:space="preserve">Quod ſi velit _Myops,_ ad diſtantiam itidem ZD diſtinctè cernens, <lb/>aſſignatum punctum A contemplari; </s>
  <s xml:id="echoid-s6953" xml:space="preserve">adſumpto, ut prius, liberè <lb/>puncto B, fiat CB = {2 AB x ZB/5 AB - 3 ZB}; </s>
  <s xml:id="echoid-s6954" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6955" xml:space="preserve">reliqua fiant, ut priùs.</s>
  <s xml:id="echoid-s6956" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6957" xml:space="preserve">_II._ </s>
  <s xml:id="echoid-s6958" xml:space="preserve">Dati puncti A perfectam imaginem, etiam ope lentis concave-<lb/>
<anchor type="note" xlink:label="note-0121-01a" xlink:href="note-0121-01"/>
convexæ, in datum aliud punctum Z lorginquius projicere.</s>
  <s xml:id="echoid-s6959" xml:space="preserve"/>
</p>
<div xml:id="echoid-div170" type="float" level="2" n="5">
<note position="right" xlink:label="note-0121-01" xlink:href="note-0121-01a" xml:space="preserve">Fig. 164.</note>
</div>
<p>
  <s xml:id="echoid-s6960" xml:space="preserve">Fiat AZ. </s>
  <s xml:id="echoid-s6961" xml:space="preserve">AD :</s>
  <s xml:id="echoid-s6962" xml:space="preserve">: 1 - R. </s>
  <s xml:id="echoid-s6963" xml:space="preserve">R. </s>
  <s xml:id="echoid-s6964" xml:space="preserve">item dividatur AD in C, ut ſit CD. <lb/></s>
  <s xml:id="echoid-s6965" xml:space="preserve">CA :</s>
  <s xml:id="echoid-s6966" xml:space="preserve">: 1. </s>
  <s xml:id="echoid-s6967" xml:space="preserve">R; </s>
  <s xml:id="echoid-s6968" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s6969" xml:space="preserve">centro C per D deſcribatur circulus EDF. </s>
  <s xml:id="echoid-s6970" xml:space="preserve">item <lb/>centro A, quopiam intervallo AB (minori quàm AD) deſcribatur <lb/>circulus EDF; </s>
  <s xml:id="echoid-s6971" xml:space="preserve">factum erit; </s>
  <s xml:id="echoid-s6972" xml:space="preserve">nempe lens EDF puncti A ima-<lb/>ginem in punctum Z projiciet.</s>
  <s xml:id="echoid-s6973" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6974" xml:space="preserve">Datâ CB, puncta A, Z viciſſim è propoſitis innoteſcunt.</s>
  <s xml:id="echoid-s6975" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6976" xml:space="preserve">In vitro, ſi CB = 15, erit {ZC = 9. </s>
  <s xml:id="echoid-s6977" xml:space="preserve">\\ ZB = 24.</s>
  <s xml:id="echoid-s6978" xml:space="preserve">} &amp; </s>
  <s xml:id="echoid-s6979" xml:space="preserve">{AZ = 16. </s>
  <s xml:id="echoid-s6980" xml:space="preserve">\\ AB = 40.</s>
  <s xml:id="echoid-s6981" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6982" xml:space="preserve">Itidem &amp; </s>
  <s xml:id="echoid-s6983" xml:space="preserve">hîc, perlentem EF verſus Z tendentes radii in A refrin-<lb/>guntur.</s>
  <s xml:id="echoid-s6984" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6985" xml:space="preserve">Hinc _Presbytis_ utile conficiatur _Vitrum,_ hocpacto: </s>
  <s xml:id="echoid-s6986" xml:space="preserve">Ad interval-<lb/>lum ZD hi diſtinctè videant. </s>
  <s xml:id="echoid-s6987" xml:space="preserve">Secetur ZD in A, ut ſit AD = {3/5} ZD. <lb/></s>
  <s xml:id="echoid-s6988" xml:space="preserve">item ſit CD = {5/8} AD (vel ſit CD = {3/8} ZD) centróque C per D <lb/>deſcribatur circulus EDF. </s>
  <s xml:id="echoid-s6989" xml:space="preserve">item utcunque ſumpto puncto B (citra <lb/>D nempe, verſus A) centro A per B deſcribatur circulus EBF. </s>
  <s xml:id="echoid-s6990" xml:space="preserve"><lb/>lente EF dicti _Presbyta_ punctum A diſtinctiſſimè conſpicient.</s>
  <s xml:id="echoid-s6991" xml:space="preserve">‖</s>
</p>
<p>
  <s xml:id="echoid-s6992" xml:space="preserve">Hiſce demum in cumulum adjiciatur ab amico communicatus _Modus_ <lb/>_elegans ac expeditus cujuſcunque caſûs imaginem Geometricè deſig-_ <lb/>_nandi; </s>
  <s xml:id="echoid-s6993" xml:space="preserve">ut &amp; </s>
  <s xml:id="echoid-s6994" xml:space="preserve">lentem deſcribendi, quæ imaginem in datum punctum_ <lb/>_projiciet._</s>
  <s xml:id="echoid-s6995" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s6996" xml:space="preserve">1. </s>
  <s xml:id="echoid-s6997" xml:space="preserve">Imaginem deſignare.</s>
  <s xml:id="echoid-s6998" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s6999" xml:space="preserve">E centris, &amp; </s>
  <s xml:id="echoid-s7000" xml:space="preserve">verticibus circulorum lentem conſtituentium erigan-<lb/>
<anchor type="note" xlink:label="note-0121-02a" xlink:href="note-0121-02"/>
tur ad axin perpendiculares Kj, BP, DQ, CI; </s>
  <s xml:id="echoid-s7001" xml:space="preserve">deinde per punctum <lb/>A ducatur quævis recta API ſecans BP, &amp; </s>
  <s xml:id="echoid-s7002" xml:space="preserve">CIinP, &amp; </s>
  <s xml:id="echoid-s7003" xml:space="preserve">I. </s>
  <s xml:id="echoid-s7004" xml:space="preserve">fac CI. <lb/></s>
  <s xml:id="echoid-s7005" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s7006" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s7007" xml:space="preserve">R. </s>
  <s xml:id="echoid-s7008" xml:space="preserve">agatur recta RP ſecans DQ, &amp; </s>
  <s xml:id="echoid-s7009" xml:space="preserve">K ρ in Q, &amp; </s>
  <s xml:id="echoid-s7010" xml:space="preserve">ρ; </s>
  <s xml:id="echoid-s7011" xml:space="preserve">fac <lb/>K ρ. </s>
  <s xml:id="echoid-s7012" xml:space="preserve">Kj :</s>
  <s xml:id="echoid-s7013" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s7014" xml:space="preserve">I, &amp; </s>
  <s xml:id="echoid-s7015" xml:space="preserve">agatur 1 Q, quæ producta ſecabit axin in Y, loco <lb/>imaginis quæſito.</s>
  <s xml:id="echoid-s7016" xml:space="preserve">‖</s>
</p>
<div xml:id="echoid-div171" type="float" level="2" n="6">
<note position="right" xlink:label="note-0121-02" xlink:href="note-0121-02a" xml:space="preserve">Fig. 165, <lb/>166, 167.</note>
</div>
<pb o="104" file="0122" n="122" rhead=""/>
<p style="it">
  <s xml:id="echoid-s7017" xml:space="preserve">2. </s>
  <s xml:id="echoid-s7018" xml:space="preserve">Reliquis datis, lentem deſcribere.</s>
  <s xml:id="echoid-s7019" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Fig. 168.</note>
<p>
  <s xml:id="echoid-s7020" xml:space="preserve">Sumantur ad arbitrium BY diſtantia lentis ab imagine, BD craſſi-<lb/>ties lentis, &amp; </s>
  <s xml:id="echoid-s7021" xml:space="preserve">alter circulorum lentem conſtituentium ut (in hoc ex-<lb/>emplo) anterior EBF, cujus centrum ſit C; </s>
  <s xml:id="echoid-s7022" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7023" xml:space="preserve">ad iſta puncta B, D, C <lb/>erigantur BP, DQ, CI ad axin perpendiculares. </s>
  <s xml:id="echoid-s7024" xml:space="preserve">deinde per pun-<lb/>ctum A ducatur recta quævis API ſecans BP, &amp; </s>
  <s xml:id="echoid-s7025" xml:space="preserve">CI in P, &amp; </s>
  <s xml:id="echoid-s7026" xml:space="preserve">I. </s>
  <s xml:id="echoid-s7027" xml:space="preserve">fiat <lb/>CI. </s>
  <s xml:id="echoid-s7028" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s7029" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s7030" xml:space="preserve">R; </s>
  <s xml:id="echoid-s7031" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7032" xml:space="preserve">agatur RP ſecans DQ in Q. </s>
  <s xml:id="echoid-s7033" xml:space="preserve">fiat DQ. </s>
  <s xml:id="echoid-s7034" xml:space="preserve">DS :</s>
  <s xml:id="echoid-s7035" xml:space="preserve">: <lb/>I. </s>
  <s xml:id="echoid-s7036" xml:space="preserve">R; </s>
  <s xml:id="echoid-s7037" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7038" xml:space="preserve">agatur SY ſecans RP productam in ρ; </s>
  <s xml:id="echoid-s7039" xml:space="preserve">à quo demittatur <lb/>perpendicularis ρ K; </s>
  <s xml:id="echoid-s7040" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7041" xml:space="preserve">centro K intervallo DK deſeribatur circulus <lb/>EDF; </s>
  <s xml:id="echoid-s7042" xml:space="preserve">erit EBFD lens quæſita.</s>
  <s xml:id="echoid-s7043" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7044" xml:space="preserve">Hîc autem in nimium excreſcenti ſpatium Lectioni defigatur limes.</s>
  <s xml:id="echoid-s7045" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div173" type="section" level="1" n="21">
<head xml:id="echoid-head24" xml:space="preserve"><emph style="sc">Lect.</emph> XV.</head>
<p>
  <s xml:id="echoid-s7046" xml:space="preserve">BEne longo circa lucis reflectiones, quatenus hæ viſum afficiunt, <lb/>inſtituto ſtadio metam nunc opportunè ſixuri videmur, ea quomo-<lb/>docunque proſecuti, quæ πρ{ou}ρμαί{te}ρα nobis viſa, nec adeò pervulgata <lb/>ſe objecerant. </s>
  <s xml:id="echoid-s7047" xml:space="preserve">quod autem magnitudines objectas attinet (quas utique <lb/>de punctis tantùm radiantibus agentes omnino videamur omiſiſie) <lb/>quales nimirum illæ ex hujuſmodi radiorum inflectionibus quoad ſi-<lb/>tum, figuram, quantitatem mutationes ſubeunt, id fermè totum paſſim <lb/>atque fuſiùs tractatum proſtat, nec animus eſt mihi toties actum agere, <lb/>vel è trivio petita quæque huc transferre. </s>
  <s xml:id="echoid-s7048" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s7049" xml:space="preserve">eò ſpectantia plera-<lb/>que cuncta de jam definitis ac oftenſis haud difficili negotio colligi <lb/>poſſe videntur; </s>
  <s xml:id="echoid-s7050" xml:space="preserve">ſingulorum nempe cujuſvis objecti punctorum (ex-<lb/>tremorum præſertim ac mediorum) apparentias indè determinando. <lb/></s>
  <s xml:id="echoid-s7051" xml:space="preserve">verùm nec ea penitus neglectui habita, ad ſubſequentem quoque <lb/>regulam (ſeu monitiunculam) preſſiùs animum advertentes forſan <lb/>autumabitis. </s>
  <s xml:id="echoid-s7052" xml:space="preserve">Si qualem aſſignata quævis ſuperficies inflectens (ſim-<lb/>plex aut compoſita) magnltudinis cujuſvis expoſitæ ſpeciem exhibet <lb/>(ampliorem nempe vel contractiorem, directam aut inverſam, confuſam <lb/>diſtinctámve, ſeu quovis alio modo demutatam) internoſcere cupiatis,
<pb o="105" file="0123" n="123" rhead=""/>
id quadantenus hoc modo pertentantes attingetis. </s>
  <s xml:id="echoid-s7053" xml:space="preserve">Oculi centrum <lb/>(quale dari paſſim ſupponitur, ei ſaltem analogum quid dari videtur; <lb/></s>
  <s xml:id="echoid-s7054" xml:space="preserve">nec indè, quoad illam quæ præ manibus rem, erroris quicquam <lb/>proveniet) oculi centrum, inquam, ubicunque pro libitu conſtitu-<lb/>tum ceu punctum radians concipiatur; </s>
  <s xml:id="echoid-s7055" xml:space="preserve">tum ex eo duo prodeuntes <lb/>radii ad propoſitam ſuperficiem (eo quem hujus exigit natura vel pro-<lb/>prietas ſpecialis modo) inflectantur. </s>
  <s xml:id="echoid-s7056" xml:space="preserve">tum inter hos inflexos colloca-<lb/>tum intelligatur objectum; </s>
  <s xml:id="echoid-s7057" xml:space="preserve">ejus certè ſpecies inter duos primos ab <lb/>oculi centro procedentes radios conſiſtet, quæ cum ipſo (quoad ap-<lb/>parentem anguli quantitatem, punctorum correſpondentium poſitionem, <lb/>&amp; </s>
  <s xml:id="echoid-s7058" xml:space="preserve">reliquas affectiones) objecto comparata voti compotes vos reddet; </s>
  <s xml:id="echoid-s7059" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s7060" xml:space="preserve">id quidem perfectius, ſi extremorum ac mediorum præſertim <lb/>objecti punctorum juſtas imagines, ex doctrina hactenus tradita, <lb/>velitis inveſtigare. </s>
  <s xml:id="echoid-s7061" xml:space="preserve">ab appoſitis exemplis res manifeſtior evadet; </s>
  <s xml:id="echoid-s7062" xml:space="preserve">in <lb/>quibus notetur punctum O ſemper oculi centrum, rectam OBA ra-<lb/>diationis axem (ſuperficiebus inflectentibus perpendicularem, &amp; </s>
  <s xml:id="echoid-s7063" xml:space="preserve">objecta <lb/>in partes æ quales dirimentem) denotare.</s>
  <s xml:id="echoid-s7064" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7065" xml:space="preserve">_Exemp. </s>
  <s xml:id="echoid-s7066" xml:space="preserve">1. </s>
  <s xml:id="echoid-s7067" xml:space="preserve">Proponatur Superficies plana medii refringentis den-_ <lb/>
<anchor type="note" xlink:label="note-0123-01a" xlink:href="note-0123-01"/>
_ſioris (aquæ ſi placet, aut vitri) objectum continentis,_ veluti Super-<lb/>ficies a recta MN repræſentata. </s>
  <s xml:id="echoid-s7068" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7069" xml:space="preserve">ab oculi centro O prodeant utcun-<lb/>que duo radii OM, ON; </s>
  <s xml:id="echoid-s7070" xml:space="preserve">qui in MF, NG refringantur; </s>
  <s xml:id="echoid-s7071" xml:space="preserve">inter <lb/>hos jam deſignetur objectum FAG (ab axe OA biſectum) hujus è <lb/>medio FGMN ſpectati ſpecies (vel apparentia) alicubi conſiſtet <lb/>inter rectas OM, ON, veluti puta ad φαγ. </s>
  <s xml:id="echoid-s7072" xml:space="preserve">cùm autem (ut ex hu-<lb/>juſce ſuperficiei natura, communíque refractionum lege palàm eſt) <lb/>ſit angulus φ O γ major angulo FOG; </s>
  <s xml:id="echoid-s7073" xml:space="preserve">hæc objecti ſpeciem ampli-<lb/>ficat inflectio. </s>
  <s xml:id="echoid-s7074" xml:space="preserve">item cùm puncta (ſibi reſpondentia) F, φ; </s>
  <s xml:id="echoid-s7075" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7076" xml:space="preserve">G, γ ad <lb/>eaſdem reſpectivè partes jaceant, ab eadem objecti poſito non immu-<lb/>tatur. </s>
  <s xml:id="echoid-s7077" xml:space="preserve">quòd ſi punctorum φ, α, γ poſitio juxta ſuperiorem doctrinam <lb/>ſtrictiùs exquiratur, de totiu<emph style="sub">s</emph> imaginis φαγ figurâ diſtantiâque <lb/>ſatìs accuratum feretur judicium.</s>
  <s xml:id="echoid-s7078" xml:space="preserve"/>
</p>
<div xml:id="echoid-div173" type="float" level="2" n="1">
<note position="right" xlink:label="note-0123-01" xlink:href="note-0123-01a" xml:space="preserve">Fig. 169.</note>
</div>
<p>
  <s xml:id="echoid-s7079" xml:space="preserve">_Exemp. </s>
  <s xml:id="echoid-s7080" xml:space="preserve">II. </s>
  <s xml:id="echoid-s7081" xml:space="preserve">Proponatur corpus denſum_ PMNQ, _Superficiebus_ <lb/>
<anchor type="note" xlink:label="note-0123-02a" xlink:href="note-0123-02"/>
_planis parallelis_ (MN, PQ) _comprehenſum;_ </s>
  <s xml:id="echoid-s7082" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7083" xml:space="preserve">ab oculi centro O <lb/>prodeuntes radii OM, ON ad ſuperficiem MN refringantur in MP, <lb/>NQ; </s>
  <s xml:id="echoid-s7084" xml:space="preserve">horum verò ad Superficiem PQ refracti ſint PF, QG (qui, <lb/>propter incidentias (ad M, P, &amp; </s>
  <s xml:id="echoid-s7085" xml:space="preserve">N, Q) pares, ipſis OM, ON <lb/>æquidiſtabunt) inter PF, QG ſtatuatur objectum FAG, cujus ſit <lb/>imago φ α γ; </s>
  <s xml:id="echoid-s7086" xml:space="preserve">tum verò manifeſtum eſt hìc ſe rem ſimiliter habere ac <lb/>in Exemplo præcedenti.</s>
  <s xml:id="echoid-s7087" xml:space="preserve"/>
</p>
<div xml:id="echoid-div174" type="float" level="2" n="2">
<note position="right" xlink:label="note-0123-02" xlink:href="note-0123-02a" xml:space="preserve">Fig. 170.</note>
</div>
<pb o="106" file="0124" n="124" rhead=""/>
<p>
  <s xml:id="echoid-s7088" xml:space="preserve">_Exemp._ </s>
  <s xml:id="echoid-s7089" xml:space="preserve">III. </s>
  <s xml:id="echoid-s7090" xml:space="preserve">_Proponatur circulus ſpecularis concavus_ M B N, &amp; </s>
  <s xml:id="echoid-s7091" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0124-01a" xlink:href="note-0124-01"/>
radiorum OM, ON reflexi ſint MF, NG (ſe decuſſantes in H, &amp; </s>
  <s xml:id="echoid-s7092" xml:space="preserve"><lb/>cum ipſis OM, ON concurrentes punctis X, Y) inter hos collocetur <lb/>objectum FAG; </s>
  <s xml:id="echoid-s7093" xml:space="preserve">ejus itidem imago rectis OM, ON interjacebit, <lb/>puta ad φαγ. </s>
  <s xml:id="echoid-s7094" xml:space="preserve">comparando jam angulos apparentes FOG, φ O γ, <lb/>clarè vides objecti FAG ſpeciem imminui. </s>
  <s xml:id="echoid-s7095" xml:space="preserve">item cernis puncta ſibi <lb/>reſpondentia F, φ, &amp; </s>
  <s xml:id="echoid-s7096" xml:space="preserve">G, γ ad alias ac alias partes jacere, ſeu objecti <lb/>ſitum hinc inverti. </s>
  <s xml:id="echoid-s7097" xml:space="preserve">Quòd ſi intra angulum &amp; </s>
  <s xml:id="echoid-s7098" xml:space="preserve">ſpatium X H Y ſtatui <lb/>concipiatur objectum, clarum eſt hinc ejus quidem ſpeciem ampliari, <lb/>ſed adhuc ſitum inverti. </s>
  <s xml:id="echoid-s7099" xml:space="preserve">ſin inter ipſa X Y conſiſtat objectum, ejus <lb/>itidem invertetur ſitus, at quantitas non immutabitur. </s>
  <s xml:id="echoid-s7100" xml:space="preserve">demùm ſi intra <lb/>angulum NHM conſtituatur objectum, puta R L S; </s>
  <s xml:id="echoid-s7101" xml:space="preserve">cujus imago <lb/>ſit ρλσ; </s>
  <s xml:id="echoid-s7102" xml:space="preserve">evidens eſt hujuſce ſpeciem creſcere, ſitúmque retineri.</s>
  <s xml:id="echoid-s7103" xml:space="preserve"/>
</p>
<div xml:id="echoid-div175" type="float" level="2" n="3">
<note position="left" xlink:label="note-0124-01" xlink:href="note-0124-01a" xml:space="preserve">Fig. 171.</note>
</div>
<p>
  <s xml:id="echoid-s7104" xml:space="preserve">_Exemp._ </s>
  <s xml:id="echoid-s7105" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s7106" xml:space="preserve">_Proponatur circulus Specularis convexus_ MBN; <lb/></s>
  <s xml:id="echoid-s7107" xml:space="preserve">
<anchor type="note" xlink:label="note-0124-02a" xlink:href="note-0124-02"/>
factiſque ſimiliter ac in eo quod immediatè præceſſit omnibus; </s>
  <s xml:id="echoid-s7108" xml:space="preserve">nè <lb/>plura prodigam verba, vides objecti F A G ſpeciem. </s>
  <s xml:id="echoid-s7109" xml:space="preserve">(φαγ) co-<lb/>arctari, ſed ejuſce poſitionem eandem perſiſtere.</s>
  <s xml:id="echoid-s7110" xml:space="preserve"/>
</p>
<div xml:id="echoid-div176" type="float" level="2" n="4">
<note position="left" xlink:label="note-0124-02" xlink:href="note-0124-02a" xml:space="preserve">Fig. 172.</note>
</div>
<p>
  <s xml:id="echoid-s7111" xml:space="preserve">_Exemp._ </s>
  <s xml:id="echoid-s7112" xml:space="preserve">V. </s>
  <s xml:id="echoid-s7113" xml:space="preserve">_Proponatur lens aliqua (exempli gratiâ, lens plano-_ <lb/>
<anchor type="note" xlink:label="note-0124-03a" xlink:href="note-0124-03"/>
_convexa)_ MBNQP. </s>
  <s xml:id="echoid-s7114" xml:space="preserve">Radii OM, ON ad ſuperficiem MBN <lb/>refringantur in MP, NQ; </s>
  <s xml:id="echoid-s7115" xml:space="preserve">tum ipſi MP, NQ ad ſuperficiem PQ <lb/>refringantur in ipſos PF, QG (ſeſe decuſſantes in H, &amp; </s>
  <s xml:id="echoid-s7116" xml:space="preserve">cum ipſis <lb/>OM, ON concurrentes ad X, Y) vides jam in prima ſigura, ſi ob-<lb/>jectum FAG infra XY (verſus H) ſtatuatur, ipſum ab imagine <lb/>φαγ majus, quàm obtutu ſimplice, repræſentari. </s>
  <s xml:id="echoid-s7117" xml:space="preserve">Quòd ſi inter ipſa <lb/>puncta X, Y ſubintelligatur collocatum, ejus quantitas neutiquam immu-<lb/>tabitur. </s>
  <s xml:id="echoid-s7118" xml:space="preserve">at ſi ſupra XY ſtatuatur objectum RLS, ejus ſpecies, ad ρ λ σ <lb/>conſpicua, diminuetur; </s>
  <s xml:id="echoid-s7119" xml:space="preserve">ubique verò punctorum correſpondentium <lb/>poſitio directa permanebit.</s>
  <s xml:id="echoid-s7120" xml:space="preserve"/>
</p>
<div xml:id="echoid-div177" type="float" level="2" n="5">
<note position="left" xlink:label="note-0124-03" xlink:href="note-0124-03a" xml:space="preserve">Fig. 173.</note>
</div>
<p>
  <s xml:id="echoid-s7121" xml:space="preserve">In altera verò figura (ubi refracti PF, QG verſus axem procur-<lb/>
<anchor type="note" xlink:label="note-0124-04a" xlink:href="note-0124-04"/>
rentes convergunt) cùm objectum FAG citra punctum H ſumitur, <lb/>vides ejus ſpeciem quantitate adauctam, at ſitu non mutatam. </s>
  <s xml:id="echoid-s7122" xml:space="preserve">verùm <lb/>objecti _RLS_ ultra concurſum H poſiti imago ρλσ nedum protorypo <lb/>major eſt, at quoad ſitum etîam eidem in verſa.</s>
  <s xml:id="echoid-s7123" xml:space="preserve"/>
</p>
<div xml:id="echoid-div178" type="float" level="2" n="6">
<note position="left" xlink:label="note-0124-04" xlink:href="note-0124-04a" xml:space="preserve">Fig. 174.</note>
</div>
<p>
  <s xml:id="echoid-s7124" xml:space="preserve">Et hoc quidem pacto nulla non lens pro varia vel objecti vel oculi <lb/>poſitione, objecti ſpeciem aliam exhibet ac aliam; </s>
  <s xml:id="echoid-s7125" xml:space="preserve">nunc dilatat, tunc <lb/>contrahit; </s>
  <s xml:id="echoid-s7126" xml:space="preserve">modò rectam dat, mox inverſam; </s>
  <s xml:id="echoid-s7127" xml:space="preserve">ſubinde propiùs adducit, <lb/>nonnunquam longiùs amovet. </s>
  <s xml:id="echoid-s7128" xml:space="preserve">Singulos caſus ad examen facilè rediges <lb/>hoc ad ſpecimen aciem mentis intendendo.</s>
  <s xml:id="echoid-s7129" xml:space="preserve"/>
</p>
<pb o="107" file="0125" n="125" rhead=""/>
<p>
  <s xml:id="echoid-s7130" xml:space="preserve">Quinimò methodum hanc leviculam adhibendo pleraſque ſuperfi-<lb/>cierum quarumvis inſlectentium hujus generis affectiones (illas nempe <lb/>quæ magnitudinum apparentes quantitates, poſitiones, diſtantias, <lb/>figuras reſpiciunt) compluriúmque _Phænomenωv_ cauſas ipſe ſtatim o-<lb/>perâ levi deprehendes; </s>
  <s xml:id="echoid-s7131" xml:space="preserve">quibus in expreſſiùs deducendis libri plures ad <lb/>tantam molem extumeſcere vel poſſunt, vel ſolent; </s>
  <s xml:id="echoid-s7132" xml:space="preserve">ut mihi ſaltem <lb/>opus non ſit hujuſmodi plura congerere. </s>
  <s xml:id="echoid-s7133" xml:space="preserve">veruntamen nè pars hæc <lb/>nimium deficiat, &amp; </s>
  <s xml:id="echoid-s7134" xml:space="preserve">quoniam nonnulla ſuccurrunt animadverſione non <lb/>indigna, de magnitudinum etiam apparentiis, tam _Dioptricis_ quàm <lb/>_Catoptricis,_ ſpecialia quædam proponam; </s>
  <s xml:id="echoid-s7135" xml:space="preserve">ea verò commodius ſe-<lb/>quentem præſtolabuntur Lectionem.</s>
  <s xml:id="echoid-s7136" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7137" xml:space="preserve">Huic interim, nè abnormiter curta ſit, aliquatenus explendæ _Pro-_ <lb/>_blemation_ hoc adnectam:</s>
  <s xml:id="echoid-s7138" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s7139" xml:space="preserve">Exponatur oculo, cujus centrum O, longinquum objectum FG, <lb/>ab oculi, circulique refringentis axe ABO biſectum; </s>
  <s xml:id="echoid-s7140" xml:space="preserve">datúſque ſit <lb/>angulus ſimpliciter (oculo nempe nudo) apparens FOG. </s>
  <s xml:id="echoid-s7141" xml:space="preserve">item aſſig-<lb/>netur punctum Z, quod imago ſit puncti A à circulo refringeute facta; <lb/></s>
  <s xml:id="echoid-s7142" xml:space="preserve">datus ſit denuò ex refractione apparens angulus POQ; </s>
  <s xml:id="echoid-s7143" xml:space="preserve">propoſitum eſt <lb/>
<anchor type="note" xlink:label="note-0125-01a" xlink:href="note-0125-01"/>
circulum iſtum refringentem deſcribere (vel determinare).</s>
  <s xml:id="echoid-s7144" xml:space="preserve"/>
</p>
<div xml:id="echoid-div179" type="float" level="2" n="7">
<note position="right" xlink:label="note-0125-01" xlink:href="note-0125-01a" xml:space="preserve">Fig. 175.</note>
</div>
<p>
  <s xml:id="echoid-s7145" xml:space="preserve">_Analyſis._ </s>
  <s xml:id="echoid-s7146" xml:space="preserve">Factum eſto; </s>
  <s xml:id="echoid-s7147" xml:space="preserve">ſit nempe circulus BN, qualis requiritur, <lb/>cujus ſit centrum C, vertex B; </s>
  <s xml:id="echoid-s7148" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7149" xml:space="preserve">qui rectam OP in N ſecet. </s>
  <s xml:id="echoid-s7150" xml:space="preserve">duca-<lb/>tur CY ad OF parallela, rectæque OP occurrens in Y, &amp; </s>
  <s xml:id="echoid-s7151" xml:space="preserve">con-<lb/>nectatur CN. </s>
  <s xml:id="echoid-s7152" xml:space="preserve">cum itaque ſit NY refractus radii ad FO, vel CY <lb/>paralleli; </s>
  <s xml:id="echoid-s7153" xml:space="preserve">erit CY. </s>
  <s xml:id="echoid-s7154" xml:space="preserve">YN :</s>
  <s xml:id="echoid-s7155" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s7156" xml:space="preserve">I. </s>
  <s xml:id="echoid-s7157" xml:space="preserve">ergò ratio CY ad YN datur; </s>
  <s xml:id="echoid-s7158" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7159" xml:space="preserve"><lb/>cùm prætereà angulus Y (dato FOP æqualis) detur, etiam (in tri-<lb/>angulo CYN) angulus CNY innoteſcet. </s>
  <s xml:id="echoid-s7160" xml:space="preserve">itaque triangulum CON <lb/>ſpecie datur; </s>
  <s xml:id="echoid-s7161" xml:space="preserve">unde ratio CO ad CN (vel CB) datur. </s>
  <s xml:id="echoid-s7162" xml:space="preserve">eſt autem <lb/>CB. </s>
  <s xml:id="echoid-s7163" xml:space="preserve">CZ :</s>
  <s xml:id="echoid-s7164" xml:space="preserve">: I - R. </s>
  <s xml:id="echoid-s7165" xml:space="preserve">R ergò ratio CB ad CZ datur. </s>
  <s xml:id="echoid-s7166" xml:space="preserve">itaque ratio <lb/>CO ad CZ quoque datur; </s>
  <s xml:id="echoid-s7167" xml:space="preserve">unde ratio CO ad OZ datur. </s>
  <s xml:id="echoid-s7168" xml:space="preserve">verùm <lb/>OZ datur; </s>
  <s xml:id="echoid-s7169" xml:space="preserve">ergò etiam CO datur. </s>
  <s xml:id="echoid-s7170" xml:space="preserve">hinc demùm &amp; </s>
  <s xml:id="echoid-s7171" xml:space="preserve">ipſa CB datur.</s>
  <s xml:id="echoid-s7172" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7173" xml:space="preserve">Componitur autem in hunc modum. </s>
  <s xml:id="echoid-s7174" xml:space="preserve">In OF utcunque capiatur <lb/>O ρ, &amp; </s>
  <s xml:id="echoid-s7175" xml:space="preserve">fiat O ρ. </s>
  <s xml:id="echoid-s7176" xml:space="preserve">O σ :</s>
  <s xml:id="echoid-s7177" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s7178" xml:space="preserve">I. </s>
  <s xml:id="echoid-s7179" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7180" xml:space="preserve">connectatur σ ρ ζ; </s>
  <s xml:id="echoid-s7181" xml:space="preserve">ducatúrque <lb/>ZRS ad ζ σ parallela. </s>
  <s xml:id="echoid-s7182" xml:space="preserve">tum fiat OZ. </s>
  <s xml:id="echoid-s7183" xml:space="preserve">ZT :</s>
  <s xml:id="echoid-s7184" xml:space="preserve">: I - R. </s>
  <s xml:id="echoid-s7185" xml:space="preserve">R (unde com-<lb/>ponendo OT. </s>
  <s xml:id="echoid-s7186" xml:space="preserve">ZT :</s>
  <s xml:id="echoid-s7187" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s7188" xml:space="preserve">R) item V = √ ZT x ZS; </s>
  <s xml:id="echoid-s7189" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7190" xml:space="preserve">X = <lb/>√ OZq - Vq; </s>
  <s xml:id="echoid-s7191" xml:space="preserve">tum X. </s>
  <s xml:id="echoid-s7192" xml:space="preserve">OZ :</s>
  <s xml:id="echoid-s7193" xml:space="preserve">: OZ. </s>
  <s xml:id="echoid-s7194" xml:space="preserve">Y. </s>
  <s xml:id="echoid-s7195" xml:space="preserve">denique X. </s>
  <s xml:id="echoid-s7196" xml:space="preserve">Y :</s>
  <s xml:id="echoid-s7197" xml:space="preserve">: OZ. <lb/></s>
  <s xml:id="echoid-s7198" xml:space="preserve">OC (unde erit Xq. </s>
  <s xml:id="echoid-s7199" xml:space="preserve">OZq :</s>
  <s xml:id="echoid-s7200" xml:space="preserve">: OZ. </s>
  <s xml:id="echoid-s7201" xml:space="preserve">OC; </s>
  <s xml:id="echoid-s7202" xml:space="preserve">hoc eſt OZq - Vq. </s>
  <s xml:id="echoid-s7203" xml:space="preserve"><lb/>OZq :</s>
  <s xml:id="echoid-s7204" xml:space="preserve">: OZ. </s>
  <s xml:id="echoid-s7205" xml:space="preserve">OC; </s>
  <s xml:id="echoid-s7206" xml:space="preserve">hoc eſt OZq - ZT x ZS. </s>
  <s xml:id="echoid-s7207" xml:space="preserve">OZq :</s>
  <s xml:id="echoid-s7208" xml:space="preserve">: OZ. </s>
  <s xml:id="echoid-s7209" xml:space="preserve"><lb/>OC). </s>
  <s xml:id="echoid-s7210" xml:space="preserve">per C verò ducatur CN ad ZS parallela, ſecans OP in N. </s>
  <s xml:id="echoid-s7211" xml:space="preserve"><lb/>denique centro C per N ducatur circulus BN; </s>
  <s xml:id="echoid-s7212" xml:space="preserve">is propoſito ſatis-<lb/>facit.</s>
  <s xml:id="echoid-s7213" xml:space="preserve"/>
</p>
<pb o="108" file="0126" n="126" rhead=""/>
<p>
  <s xml:id="echoid-s7214" xml:space="preserve">Nam ob OZq - ZT x ZS. </s>
  <s xml:id="echoid-s7215" xml:space="preserve">OZq :</s>
  <s xml:id="echoid-s7216" xml:space="preserve">: OZ. </s>
  <s xml:id="echoid-s7217" xml:space="preserve">OC; </s>
  <s xml:id="echoid-s7218" xml:space="preserve">erit OZ cub <lb/> = OC x OZq - OC x ZT x ZS. </s>
  <s xml:id="echoid-s7219" xml:space="preserve">tranſponendóque OC x ZT x ZS = OC <lb/>x OZq - OZ cub. </s>
  <s xml:id="echoid-s7220" xml:space="preserve">atqui propter OZ. </s>
  <s xml:id="echoid-s7221" xml:space="preserve">ZS :</s>
  <s xml:id="echoid-s7222" xml:space="preserve">: OC. </s>
  <s xml:id="echoid-s7223" xml:space="preserve">CN. </s>
  <s xml:id="echoid-s7224" xml:space="preserve">eſt <lb/>OZ x CN = ZS x OC. </s>
  <s xml:id="echoid-s7225" xml:space="preserve">quare OZ x CN x ZT = OC x OZq <lb/>- OZ cub; </s>
  <s xml:id="echoid-s7226" xml:space="preserve">adeóque (elidendo OZ) erit CN x ZT = OC <lb/>x OZ - OZq. </s>
  <s xml:id="echoid-s7227" xml:space="preserve">vel CN. </s>
  <s xml:id="echoid-s7228" xml:space="preserve">OC - OZ:</s>
  <s xml:id="echoid-s7229" xml:space="preserve">: OZ. </s>
  <s xml:id="echoid-s7230" xml:space="preserve">ZT; </s>
  <s xml:id="echoid-s7231" xml:space="preserve">hoc eſt CB. </s>
  <s xml:id="echoid-s7232" xml:space="preserve">CZ:</s>
  <s xml:id="echoid-s7233" xml:space="preserve">: <lb/>OT. </s>
  <s xml:id="echoid-s7234" xml:space="preserve">ZT.</s>
  <s xml:id="echoid-s7235" xml:space="preserve">. &amp; </s>
  <s xml:id="echoid-s7236" xml:space="preserve">componendo BZ. </s>
  <s xml:id="echoid-s7237" xml:space="preserve">CZ :</s>
  <s xml:id="echoid-s7238" xml:space="preserve">: OT. </s>
  <s xml:id="echoid-s7239" xml:space="preserve">ZT :</s>
  <s xml:id="echoid-s7240" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s7241" xml:space="preserve">R. </s>
  <s xml:id="echoid-s7242" xml:space="preserve">itaque <lb/>primò liquet punctum Z imaginem eſſe puncti A, ex refractione <lb/>factam ad circulum BN. </s>
  <s xml:id="echoid-s7243" xml:space="preserve">quinetiam ob CY. </s>
  <s xml:id="echoid-s7244" xml:space="preserve">YN :</s>
  <s xml:id="echoid-s7245" xml:space="preserve">: ρ O. </s>
  <s xml:id="echoid-s7246" xml:space="preserve">O σ :</s>
  <s xml:id="echoid-s7247" xml:space="preserve">: <lb/>R. </s>
  <s xml:id="echoid-s7248" xml:space="preserve">I; </s>
  <s xml:id="echoid-s7249" xml:space="preserve">palàm eſt NO refractum eſſe radii ad CY, hoc eſt ad FO <lb/>paralleli. </s>
  <s xml:id="echoid-s7250" xml:space="preserve">liquidò proinde conſtat propoſitum.</s>
  <s xml:id="echoid-s7251" xml:space="preserve">‖</s>
</p>
<p>
  <s xml:id="echoid-s7252" xml:space="preserve">In hoc caſu debet eſſe OZq &amp;</s>
  <s xml:id="echoid-s7253" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7254" xml:space="preserve">ZT x ZS. </s>
  <s xml:id="echoid-s7255" xml:space="preserve">Haud abſimili ratione quoad <lb/>alios caſus (ut ſi circuli refringentis cavum objecto exponatur, &amp; </s>
  <s xml:id="echoid-s7256" xml:space="preserve">c.) <lb/></s>
  <s xml:id="echoid-s7257" xml:space="preserve">peragetur negotium. </s>
  <s xml:id="echoid-s7258" xml:space="preserve">ego ſpecimen tantùm _inſtitui Problematis,_ juxta <lb/>quod viſibilis objecti ſpecies per refractionem circularem ſecundum <lb/>præſtitutas quantitatem atque diſtantiam utcunque poſſit immutari.</s>
  <s xml:id="echoid-s7259" xml:space="preserve">‖</s>
</p>
</div>
<div xml:id="echoid-div181" type="section" level="1" n="22">
<head xml:id="echoid-head25" style="it" xml:space="preserve">APPENDICVLA.</head>
<p>
  <s xml:id="echoid-s7260" xml:space="preserve">UT hæc paullò ſtrigoſior Lectio nonnihil incraſſetur, faciam hîc <lb/>(quanquam alienore loco) quod alibi (ſi mihi tunc in mentem <lb/>veniſſet) factum oportebat; </s>
  <s xml:id="echoid-s7261" xml:space="preserve">raciociniis noſtris adverſantem, à viro <lb/>doctiſſimo (alioquin opinor rarò dormitante) commiſſum paralo-<lb/>giſmum, nè cui fraudi ſit, detegam ac amoliar; </s>
  <s xml:id="echoid-s7262" xml:space="preserve">unáque doctrinam <lb/>noſtram confirmabo. </s>
  <s xml:id="echoid-s7263" xml:space="preserve">horſum è præmiſſis conſequens, ſed &amp; </s>
  <s xml:id="echoid-s7264" xml:space="preserve">expe-<lb/>rientiæ (ut videbimus) conſonum hoc præſterno: </s>
  <s xml:id="echoid-s7265" xml:space="preserve">E refractione quavis <lb/>(nec non è reflectione ad circulum) duobus oculis apprehenſum ob-<lb/>jectum (puta lwcidum punctum A) reverà duplum apparet, ſeu duas (ad <lb/>minus) obtinet imagines.</s>
  <s xml:id="echoid-s7266" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7267" xml:space="preserve">Nam à puncto A exeuntes inſlectenti M N incidant duo quicunque <lb/>
<anchor type="note" xlink:label="note-0126-01a" xlink:href="note-0126-01"/>
radii AM, AN; </s>
  <s xml:id="echoid-s7268" xml:space="preserve">quorum inflexi ſint ME, NF; </s>
  <s xml:id="echoid-s7269" xml:space="preserve">concurrentes in X; <lb/></s>
  <s xml:id="echoid-s7270" xml:space="preserve">in his autem uſpiam conſtituantur oculorum centra O, P. </s>
  <s xml:id="echoid-s7271" xml:space="preserve">quòd puncti <lb/>A imago nulla ad occurſum X exiſtat, è ſupra poſitis, ac probatis con-<lb/>ſectatur (omnes enim imagines ad illa conſiſtere docuimus inflexorum <lb/>puncta, ad quæ nulli illos alii inflexi interſecant) itaque duæ ſunt <lb/>imagines puncti A, una in inflexo EM (qualis α) ad oculum O per-<lb/>tinens; </s>
  <s xml:id="echoid-s7272" xml:space="preserve">altera in inflexo FN (qualis α) oculo P deputanda.</s>
  <s xml:id="echoid-s7273" xml:space="preserve"/>
</p>
<div xml:id="echoid-div181" type="float" level="2" n="1">
<note position="left" xlink:label="note-0126-01" xlink:href="note-0126-01a" xml:space="preserve">Fig. 176.</note>
</div>
<p>
  <s xml:id="echoid-s7274" xml:space="preserve">Hinc liquet etiam magnitudinis cujuſvis hoc modo ſpectatæ duplicem <lb/>imaginem haberi.</s>
  <s xml:id="echoid-s7275" xml:space="preserve"/>
</p>
<pb o="109" file="0127" n="127" rhead=""/>
<p>
  <s xml:id="echoid-s7276" xml:space="preserve">Huic effato ſi contraria obtendatur experientia, monſtrans ſubinde <lb/>duntaxat unam imaginem apparere; </s>
  <s xml:id="echoid-s7277" xml:space="preserve">regero, in refractione quidem <lb/>ad ſuperficiem planam apparenter hoc plerumque contingere, quo-<lb/>niam imagines iſtæ duæ (quales α, _a_) ita ſibimet ipſis, ita refracto-<lb/>rum concurſui X vicinæ ſunt, ut ipſarum intervallum diſcerni nequeat, <lb/>ipſæque (ſicut in ſimili caſu obvenire mox oſtendemus) velut in unam <lb/>imaginem interceptibilitèr coaleſcant; </s>
  <s xml:id="echoid-s7278" xml:space="preserve">aſt in aliis diverſi generis in-<lb/>flectionibus, etiam ſenſu conteſtante, maniſeſtè ſecùs apparet; </s>
  <s xml:id="echoid-s7279" xml:space="preserve">id <lb/>quod cùm è compluribus admodum obviis experimentis conſtare poſſit, <lb/>unum ſaltem ac alterum proponemns. </s>
  <s xml:id="echoid-s7280" xml:space="preserve">Speculo BNM exponatur ob-<lb/>
<anchor type="note" xlink:label="note-0127-01a" xlink:href="note-0127-01"/>
jectum A; </s>
  <s xml:id="echoid-s7281" xml:space="preserve">tum oculis, velut ad O, P conſtitutis, apparebit ejuſce <lb/>duplex ſpecies α, _a_; </s>
  <s xml:id="echoid-s7282" xml:space="preserve">quarum illa (a) clauſo oculo O, hæc (_a_) clauſo <lb/>P diſparebit.</s>
  <s xml:id="echoid-s7283" xml:space="preserve"/>
</p>
<div xml:id="echoid-div182" type="float" level="2" n="2">
<note position="right" xlink:label="note-0127-01" xlink:href="note-0127-01a" xml:space="preserve">Fig. 177, <lb/>178.</note>
</div>
<p>
  <s xml:id="echoid-s7284" xml:space="preserve">Notetur autem, ſi placet, imaginum α, _a_ intervalla (pro vario <lb/>oculorum ſitu) nunc magìs, nunc minùs deduci, ſic ut ſubinde coadu-<lb/>nari videantur. </s>
  <s xml:id="echoid-s7285" xml:space="preserve">Nempe ſi oculus P ad F concipiatur tranſlatus, duca-<lb/>túrque FG ipſi PO parallela, &amp; </s>
  <s xml:id="echoid-s7286" xml:space="preserve">æqualis; </s>
  <s xml:id="echoid-s7287" xml:space="preserve">unde jam &amp; </s>
  <s xml:id="echoid-s7288" xml:space="preserve">oculus O <lb/>in C poſitus concipiatur; </s>
  <s xml:id="echoid-s7289" xml:space="preserve">quoniam FE minor eſt quàm FG, radius <lb/>MO per G non tranſibit; </s>
  <s xml:id="echoid-s7290" xml:space="preserve">tranſeat alter inflexus LG; </s>
  <s xml:id="echoid-s7291" xml:space="preserve">in hoc itaque <lb/>jam conſiſtet imago α, ab altera _a_; </s>
  <s xml:id="echoid-s7292" xml:space="preserve">magìs elongata. </s>
  <s xml:id="echoid-s7293" xml:space="preserve">Reliquarum hujuſ-<lb/>modi diverſitatum haud diſpar aſſignari poterit ratio.</s>
  <s xml:id="echoid-s7294" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7295" xml:space="preserve">Adjungatur &amp; </s>
  <s xml:id="echoid-s7296" xml:space="preserve">hoc, an paſſim obſervatum neſcio, dignum certè <lb/>
<anchor type="note" xlink:label="note-0127-02a" xlink:href="note-0127-02"/>
quod obſervetur: </s>
  <s xml:id="echoid-s7297" xml:space="preserve">Ad ſpeculum concavum RSMN faciem tuam <lb/>FAG (ſpeculo propiùs ad motam) contemplare. </s>
  <s xml:id="echoid-s7298" xml:space="preserve">Et primò quidem <lb/>oculo O (altero P occluſo) cernes ejus imaginem φαγ; </s>
  <s xml:id="echoid-s7299" xml:space="preserve">rurſus (ocu-<lb/>lo O occluſo) altero P conſpicies imaginem fag, à priore φαγ ali-<lb/>quantùm deflectentem; </s>
  <s xml:id="echoid-s7300" xml:space="preserve">demùm utroque ſimul oculo recluſo ſpectans <lb/>itlas in unam coalitas percipies; </s>
  <s xml:id="echoid-s7301" xml:space="preserve">ſeu, ſpeciem unam aſpicies, per-<lb/>quam notabili diſcrimine, ampliorem priorum ſingularum alterutrà.</s>
  <s xml:id="echoid-s7302" xml:space="preserve"/>
</p>
<div xml:id="echoid-div183" type="float" level="2" n="3">
<note position="right" xlink:label="note-0127-02" xlink:href="note-0127-02a" xml:space="preserve">Fig 179.</note>
</div>
<p>
  <s xml:id="echoid-s7303" xml:space="preserve">Exhinc, obiter, ſuſpicari licet, etiam intuitum ſimplicem adhiben-<lb/>tibus objecta binis oculis ſpectata tantillo majora videri, quàm uno; <lb/></s>
  <s xml:id="echoid-s7304" xml:space="preserve">ſpeciebus ità coëuntibus, ut non exquiſitè congruant.</s>
  <s xml:id="echoid-s7305" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7306" xml:space="preserve">Unicam prætereà ſubdemus inſtantiam: </s>
  <s xml:id="echoid-s7307" xml:space="preserve">Per ſphæram vitream (aut <lb/>
<anchor type="note" xlink:label="note-0127-03a" xlink:href="note-0127-03"/>
ſi mavis; </s>
  <s xml:id="echoid-s7308" xml:space="preserve">per phialam conicam aut cylindricam aquâ repletam) MBN <lb/>tranſlucentem lucernulæ flammam A ſpecta; </s>
  <s xml:id="echoid-s7309" xml:space="preserve">ejus duas imagines a, _a_; <lb/></s>
  <s xml:id="echoid-s7310" xml:space="preserve">obſervabis (pro oculorum ſitu magìs à ſe minúſve diſſitas) quarum <lb/>una (a) clauſo oculo O, altera (_a_) clauſo P evaneſcet.</s>
  <s xml:id="echoid-s7311" xml:space="preserve"/>
</p>
<div xml:id="echoid-div184" type="float" level="2" n="4">
<note position="right" xlink:label="note-0127-03" xlink:href="note-0127-03a" xml:space="preserve">Fig 180.</note>
</div>
<p>
  <s xml:id="echoid-s7312" xml:space="preserve">Videtur hæc inſtantia vel ſola ſufficere vulgari ſententiæ refellendæ; <lb/></s>
  <s xml:id="echoid-s7313" xml:space="preserve">
<anchor type="note" xlink:label="note-0127-04a" xlink:href="note-0127-04"/>
juxta quam (ut _Keplerus_ alicubi colligit) puncti A ſimplex imago <lb/>ad punctum X conſiſteret.</s>
  <s xml:id="echoid-s7314" xml:space="preserve"/>
</p>
<div xml:id="echoid-div185" type="float" level="2" n="5">
<note position="right" xlink:label="note-0127-04" xlink:href="note-0127-04a" xml:space="preserve">_Paralipom. pag._ <lb/>178.</note>
</div>
<pb o="110" file="0128" n="128" rhead=""/>
<p>
  <s xml:id="echoid-s7315" xml:space="preserve">Has inſtantias, facilitatis gratiâ, ità propoſuimus, quaſi punctum <lb/>A, unà cum duobus oculis O, P in plano exiſteret ad ſuperficiem in-<lb/>flectentem recto. </s>
  <s xml:id="echoid-s7316" xml:space="preserve">id quod utrùm in experiendo præcisè contingat <lb/>nécne, parùm refert; </s>
  <s xml:id="echoid-s7317" xml:space="preserve">duas utcunque ſpecies apparere liquet. </s>
  <s xml:id="echoid-s7318" xml:space="preserve">quin <lb/>facilè concipitur etiam eo poſito rem non aliter ſe habituram.</s>
  <s xml:id="echoid-s7319" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7320" xml:space="preserve">His prælibatis, illud diſcutiamus, quod innuimus, ψευ{δ ο}{γ ρ}άφημæ <lb/>
<anchor type="note" xlink:label="note-0128-01a" xlink:href="note-0128-01"/>
quo nempe P. </s>
  <s xml:id="echoid-s7321" xml:space="preserve">_Herigonius_ propoſitionem hanc ſuam comprobatum it: <lb/></s>
  <s xml:id="echoid-s7322" xml:space="preserve">“ Si oculus, &amp; </s>
  <s xml:id="echoid-s7323" xml:space="preserve">aſpectabile ſint in diverſis mediis ſè mutuo contingen-<lb/>tibus, imago apparebit in concurſu catheti, &amp; </s>
  <s xml:id="echoid-s7324" xml:space="preserve">radii ab oculo per <lb/>punctum refractionis directè producti.</s>
  <s xml:id="echoid-s7325" xml:space="preserve"/>
</p>
<div xml:id="echoid-div186" type="float" level="2" n="6">
<note position="left" xlink:label="note-0128-01" xlink:href="note-0128-01a" xml:space="preserve">_In Dioper._</note>
</div>
<p>
  <s xml:id="echoid-s7326" xml:space="preserve">Sit utique punctum Fin medio denſiori HLMN collocatum, quod <lb/>
<anchor type="note" xlink:label="note-0128-02a" xlink:href="note-0128-02"/>
ad oculos A, B radios FEA, FDB emittat refractos ad E, &amp; </s>
  <s xml:id="echoid-s7327" xml:space="preserve">D; <lb/></s>
  <s xml:id="echoid-s7328" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7329" xml:space="preserve">rectæ AE, BD conveniant in C; </s>
  <s xml:id="echoid-s7330" xml:space="preserve">ſit autem ſuperficiei refringenti <lb/>perpendicularis recta FG; </s>
  <s xml:id="echoid-s7331" xml:space="preserve">erit (inquit) puncti F imago in recta <lb/>FG. </s>
  <s xml:id="echoid-s7332" xml:space="preserve">id quod ità demonſtrat: </s>
  <s xml:id="echoid-s7333" xml:space="preserve">Quoniam dicta imago tam in refracto <lb/>AE, quàm in refracto BD exiftit, ergò in horum interſectione C <lb/>exiſtet. </s>
  <s xml:id="echoid-s7334" xml:space="preserve">verùm interſectio C ìn recta FG exiſtet; </s>
  <s xml:id="echoid-s7335" xml:space="preserve">quoniam hæc <lb/>communis eſt ſectio planorum AEF, BDF ſuperficiei refringenti <lb/>rectorum. </s>
  <s xml:id="echoid-s7336" xml:space="preserve">ergò liquet propoſitum.</s>
  <s xml:id="echoid-s7337" xml:space="preserve"/>
</p>
<div xml:id="echoid-div187" type="float" level="2" n="7">
<note position="left" xlink:label="note-0128-02" xlink:href="note-0128-02a" xml:space="preserve">Fig. 181.</note>
</div>
<p>
  <s xml:id="echoid-s7338" xml:space="preserve">In hanc demonſtrationem adverto; </s>
  <s xml:id="echoid-s7339" xml:space="preserve">1. </s>
  <s xml:id="echoid-s7340" xml:space="preserve">Supponit ea refractos AE, <lb/>BD concurrere; </s>
  <s xml:id="echoid-s7341" xml:space="preserve">quod tamen falſum eſt, præterquam in uno vel al-<lb/>tero caſu; </s>
  <s xml:id="echoid-s7342" xml:space="preserve">quum nempe planum ABF in eodem exiſtit cum ipſa <lb/>recta FG plano; </s>
  <s xml:id="echoid-s7343" xml:space="preserve">vel, cùm puncta A, B ſunt in ſuperficie coni recti, <lb/>cujus axis eſt recta FG. </s>
  <s xml:id="echoid-s7344" xml:space="preserve">quod ſi prior caſus ponatur, è ſuprà demon-<lb/>ſtratis maniſeſtum eſt refractos AE, BD non in recta FG, ſed intra <lb/>angulum FGH convenire; </s>
  <s xml:id="echoid-s7345" xml:space="preserve">quod è principiis noſtris elicitum illum <lb/>ſaltem conſtringere debet, qui principia iſta admittit ac amplectitur. <lb/></s>
  <s xml:id="echoid-s7346" xml:space="preserve">2. </s>
  <s xml:id="echoid-s7347" xml:space="preserve">Hinc, illa demonſtratio ipſam ſe perimit: </s>
  <s xml:id="echoid-s7348" xml:space="preserve">Nam, quoniam (in po-<lb/>ſito caſu) puncti F imago tam in recta AE, quàm in recta BD exiſtit, <lb/>adeóque in harum concurſu; </s>
  <s xml:id="echoid-s7349" xml:space="preserve">concurſus autem iſte non eſt in recta <lb/>FG; </s>
  <s xml:id="echoid-s7350" xml:space="preserve">ergò liquet dictam imaginem extra rectam FG verſari. </s>
  <s xml:id="echoid-s7351" xml:space="preserve">3. </s>
  <s xml:id="echoid-s7352" xml:space="preserve">Sup-<lb/>ponit iſte diſcurſus (ut &amp; </s>
  <s xml:id="echoid-s7353" xml:space="preserve">ſuppar ille jamjam prolatus) puncti F uni-<lb/>cam oculo u@ique imaginem apparere; </s>
  <s xml:id="echoid-s7354" xml:space="preserve">quod πρῶγον ψεῦδ{ος} erat, <lb/>à nobis paullo ſupra refutatum. </s>
  <s xml:id="echoid-s7355" xml:space="preserve">Enimverò diverſi oculi ſunt reipsâ <lb/>diverſi ſpectatores. </s>
  <s xml:id="echoid-s7356" xml:space="preserve">hæc, opinor, ratiocinium illud ſatìs enervant.</s>
  <s xml:id="echoid-s7357" xml:space="preserve">‖</s>
</p>
<pb o="111" file="0129" n="129"/>
</div>
<div xml:id="echoid-div189" type="section" level="1" n="23">
<head xml:id="echoid-head26" xml:space="preserve"><emph style="sc">Lect.</emph> XVI.</head>
<p>
  <s xml:id="echoid-s7358" xml:space="preserve">I. </s>
  <s xml:id="echoid-s7359" xml:space="preserve">P_Unctorum ex inflectione determinatis apparentibus locis,_ con-<lb/>quieſcere poſſem; </s>
  <s xml:id="echoid-s7360" xml:space="preserve">ſiquidem exinde magnitudinum apparentiæ <lb/>deducuntur, quotlibet in ipſis exiſtentium punctorum imagines de-<lb/>ſignando. </s>
  <s xml:id="echoid-s7361" xml:space="preserve">cæterùm nè juſto parciùs in hac parte, vel illiberaliùs egiſſe <lb/>videar, etiam _de rectarum linearum (conſequenter &amp; </s>
  <s xml:id="echoid-s7362" xml:space="preserve">planarum ſuper-_ <lb/>_ficierum, quibus diſtinctè viſui repræſentandis natur a præcipuè conſu-_ <lb/>_luiſſe videtur) apparentiis &amp; </s>
  <s xml:id="echoid-s7363" xml:space="preserve">imaginibus expreſſiora ſpecimina quædam_ <lb/>_haud gravabor adnectere._ </s>
  <s xml:id="echoid-s7364" xml:space="preserve">de quibus etiam circa reliquarum magnitu-<lb/>dinum apparentias propius ac promptius fiat judicium.</s>
  <s xml:id="echoid-s7365" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7366" xml:space="preserve">II. </s>
  <s xml:id="echoid-s7367" xml:space="preserve">Notetur autem imprimìs; </s>
  <s xml:id="echoid-s7368" xml:space="preserve">Sicuti (quod ſæpiùs in antedictis <lb/>habetur inſinuatum) cujuſque puncti quodammodò duplex eſt imago; <lb/></s>
  <s xml:id="echoid-s7369" xml:space="preserve">una ſimplex, abſoluta, principalis; </s>
  <s xml:id="echoid-s7370" xml:space="preserve">illa ſcilicet, quæ in recta verſatur <lb/>ad ſuperficiem inflectentem perpendiculari, perque radians punctum <lb/>ſimul ac oculi centrum tranſeunte (hoc eſt in communi lucidæ radia-<lb/>tionis, ſuperficiei reflectentis, ipſiúſque viſionis axe) altera verò <lb/>relata, mutabilis, ac minùs præcipua; </s>
  <s xml:id="echoid-s7371" xml:space="preserve">quæ talis eſt reſpectu oculi <lb/>extra rectam inflectenti ſuperficiei perpendicularem arbitrariè conſti-<lb/>tuti; </s>
  <s xml:id="echoid-s7372" xml:space="preserve">ità pari fermè modo duplex cujuſque magnitudinis imago con-<lb/>cipi poteſt; </s>
  <s xml:id="echoid-s7373" xml:space="preserve">una quidem abſoluta (quam ſaltem hoc nomine deſig-<lb/>nabo) quæ ex punctorum ſingulorum in ipſa exiſtentium abſolutis <lb/>imaginibus quaſi conflatur, illas ſaltem comprehendit (qualis in ob-<lb/>jecta congrua ſuperficie vividè deformaretur; </s>
  <s xml:id="echoid-s7374" xml:space="preserve">qualíſque videretur <lb/>oculo ad infinitam ab inflectente ſuperficie diſtantiam ritè collocato) <lb/>altera verò relata, quæ oculum reſpicit ubivis in certa poſitione con-<lb/>ſtitutum; </s>
  <s xml:id="echoid-s7375" xml:space="preserve">quid velim, &amp; </s>
  <s xml:id="echoid-s7376" xml:space="preserve">quare ſic diſtinguam ab exemplis benè mul-<lb/>tis in decurſu proponendis luculenter apparebit.</s>
  <s xml:id="echoid-s7377" xml:space="preserve"/>
</p>
<pb o="112" file="0130" n="130" rhead=""/>
<p>
  <s xml:id="echoid-s7378" xml:space="preserve">III. </s>
  <s xml:id="echoid-s7379" xml:space="preserve">_Superſiciem planam media dirimentem (aquam ſiplacet ac aërem)_ <lb/>
<anchor type="note" xlink:label="note-0130-01a" xlink:href="note-0130-01"/>
repræſentet recta PQ, &amp; </s>
  <s xml:id="echoid-s7380" xml:space="preserve">aquæ inſit recta FP ad PQ perpen-<lb/>dicularis. </s>
  <s xml:id="echoid-s7381" xml:space="preserve">ſiat autem FP. </s>
  <s xml:id="echoid-s7382" xml:space="preserve">XP :</s>
  <s xml:id="echoid-s7383" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s7384" xml:space="preserve">I; </s>
  <s xml:id="echoid-s7385" xml:space="preserve">erit X P imago abſoluta rectæ <lb/>FP; </s>
  <s xml:id="echoid-s7386" xml:space="preserve">continet illa ſcilicet omnes locos punctorum, quæ in FP, oculo <lb/>apparentes in ipſa FP ſito. </s>
  <s xml:id="echoid-s7387" xml:space="preserve">verùm ſi ponatur oculus uſpiam extra FP, <lb/>velut ad O, ei tota FP citra XP apparebit. </s>
  <s xml:id="echoid-s7388" xml:space="preserve">tranſeat videlicet ali-<lb/>cujus radii FM refractus per O, &amp; </s>
  <s xml:id="echoid-s7389" xml:space="preserve">protrahatur OM, ut occurrat ipſi <lb/>FP in K. </s>
  <s xml:id="echoid-s7390" xml:space="preserve">eſt ergò (ſecundum præmonſtrata) punctum Kinter X, &amp; </s>
  <s xml:id="echoid-s7391" xml:space="preserve"><lb/>P. </s>
  <s xml:id="echoid-s7392" xml:space="preserve">itidem (è prius oſtenſis) puncti F imago quæ in refracto OMK, <lb/>ad oculum O relata, inter K, &amp; </s>
  <s xml:id="echoid-s7393" xml:space="preserve">M cadit, veluti puta ad φ. </s>
  <s xml:id="echoid-s7394" xml:space="preserve">ſimili ratione <lb/>cujuſvis alterius in ipſa FP accepti puncti, ceu R, imago (cogita ρ) <lb/>citra rectam XP, verſus oculum, jacet. </s>
  <s xml:id="echoid-s7395" xml:space="preserve">totius itaque rectæ FP imago <lb/>talis eſt, qualem curva linea φρ P refert. </s>
  <s xml:id="echoid-s7396" xml:space="preserve">quòd ſi P F infinitè protra-<lb/>hatur, ejus totius imago P ρρ verſus aſymptoton OBA, ad PF paralle-<lb/>lam, accedens excurrit.</s>
  <s xml:id="echoid-s7397" xml:space="preserve"/>
</p>
<div xml:id="echoid-div189" type="float" level="2" n="1">
<note position="left" xlink:label="note-0130-01" xlink:href="note-0130-01a" xml:space="preserve">Fig. 182.</note>
</div>
<p>
  <s xml:id="echoid-s7398" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s7399" xml:space="preserve">_Delineatur autem curva P ρ φ hoc modo._ </s>
  <s xml:id="echoid-s7400" xml:space="preserve">ab O ducatur utcun-<lb/>que recta OMK ſecans rectam PQ in M; </s>
  <s xml:id="echoid-s7401" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7402" xml:space="preserve">(poſito fore S = √ Rq <lb/>- Iq) ſit PH = {Sq x PM cub/Iq x FPq}; </s>
  <s xml:id="echoid-s7403" xml:space="preserve">atque per H ducatur H φ ad PR <lb/>parallela, ipſi OK occurrens in φ; </s>
  <s xml:id="echoid-s7404" xml:space="preserve">erit φ in dicta linea; </s>
  <s xml:id="echoid-s7405" xml:space="preserve">nempe, ſi <lb/>OMK ipſius M F refractus concipiatur, erit punctum φ ipſius F ima-<lb/>go. </s>
  <s xml:id="echoid-s7406" xml:space="preserve">eodem modo reliqua lineæ P ρ ρ puncta deſignantur.</s>
  <s xml:id="echoid-s7407" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7408" xml:space="preserve">V. </s>
  <s xml:id="echoid-s7409" xml:space="preserve">Quinetiam adſumptâ rectâ FG ad PQ parallelâ, ductâque <lb/>
<anchor type="note" xlink:label="note-0130-02a" xlink:href="note-0130-02"/>
GQ ad FP (vel ABO): </s>
  <s xml:id="echoid-s7410" xml:space="preserve">parallelâ; </s>
  <s xml:id="echoid-s7411" xml:space="preserve">item per X ductà X α Y ad PQ <lb/>parallelâ, erit quidem recta X α Y rectæ FAG imago abſoluta; <lb/></s>
  <s xml:id="echoid-s7412" xml:space="preserve">verùm ejus imago ad oculum O relata citra rectam XY tota jacet, eám-<lb/>que curva φαγ repræſentat, admodum jamjam præſcriptum puncta-<lb/>tim delineabilis. </s>
  <s xml:id="echoid-s7413" xml:space="preserve">itaque compoſitæ lineæ PFGQ, circa axem OBA <lb/>rotatæ, imago fornicem referet arcuatam. </s>
  <s xml:id="echoid-s7414" xml:space="preserve">id quod experiri vos velim <lb/>vaſculi cylindrici aquâ repleti ſuperficiem inſpectando.</s>
  <s xml:id="echoid-s7415" xml:space="preserve"/>
</p>
<div xml:id="echoid-div190" type="float" level="2" n="2">
<note position="left" xlink:label="note-0130-02" xlink:href="note-0130-02a" xml:space="preserve">Fig. 182.</note>
</div>
<p>
  <s xml:id="echoid-s7416" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s7417" xml:space="preserve">Quòd ſirecta viſibilis FG ad PQ inclinata ſit, cum ea conve-<lb/>
<anchor type="note" xlink:label="note-0130-03a" xlink:href="note-0130-03"/>
niens in V; </s>
  <s xml:id="echoid-s7418" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7419" xml:space="preserve">connectatur XV, erit rurſus XY ipſius FG imago ab-<lb/>ſoluta; </s>
  <s xml:id="echoid-s7420" xml:space="preserve">relatam verò curva φαγ repræſentat.</s>
  <s xml:id="echoid-s7421" xml:space="preserve"/>
</p>
<div xml:id="echoid-div191" type="float" level="2" n="3">
<note position="left" xlink:label="note-0130-03" xlink:href="note-0130-03a" xml:space="preserve">Fig. 183.</note>
</div>
<p>
  <s xml:id="echoid-s7422" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s7423" xml:space="preserve">Quòd ſi viciſſim _oculus O in aqua ponatur conſtitutus,_ &amp; </s>
  <s xml:id="echoid-s7424" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0130-04a" xlink:href="note-0130-04"/>
ab indèreſpiciatur recta P F in aëre poſita, fiátque rurſus PF. </s>
  <s xml:id="echoid-s7425" xml:space="preserve">PX :</s>
  <s xml:id="echoid-s7426" xml:space="preserve">:
<pb o="113" file="0131" n="131" rhead=""/>
R. </s>
  <s xml:id="echoid-s7427" xml:space="preserve">I; </s>
  <s xml:id="echoid-s7428" xml:space="preserve">erit quidem XP imago rectæ FP abſoluta; </s>
  <s xml:id="echoid-s7429" xml:space="preserve">at ejuſdem imago <lb/>relata (puta P ρ φ ρ) ultra PFR jacet, ab illa ſenſim reclinans; </s>
  <s xml:id="echoid-s7430" xml:space="preserve">e-<lb/>júſque puncta quælibet ità ſignantur. </s>
  <s xml:id="echoid-s7431" xml:space="preserve">Ab O ducatur recta OK ut-<lb/>cunque rectam PQ ſecans in M, &amp; </s>
  <s xml:id="echoid-s7432" xml:space="preserve">ſit KM ipſius FM refractus, <lb/>tum (poſito rurſus S = √ Iq - Rq) fiat PH = {Sq x PMq/lq x PFq} PM, <lb/>&amp; </s>
  <s xml:id="echoid-s7433" xml:space="preserve">per H ad PF parallela ducatur H φ, ipſam OMK interſecans <lb/>ad φ; </s>
  <s xml:id="echoid-s7434" xml:space="preserve">erit punctum φ in dicta linea, punctum ſcilicet F repræſentans. <lb/></s>
  <s xml:id="echoid-s7435" xml:space="preserve">eodémque modo puncta quotlibet alia deprehendes.</s>
  <s xml:id="echoid-s7436" xml:space="preserve"/>
</p>
<div xml:id="echoid-div192" type="float" level="2" n="4">
<note position="left" xlink:label="note-0130-04" xlink:href="note-0130-04a" xml:space="preserve">Fig. 184.</note>
</div>
<p>
  <s xml:id="echoid-s7437" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s7438" xml:space="preserve">Similiter rectæ FG ad ipſam PQ parallelæ, vel inclinatæ <lb/>imago relata φ α γ (in partes arcuata contrarias illis, ad quas prioris. <lb/></s>
  <s xml:id="echoid-s7439" xml:space="preserve">casûs imago videbatur incurvata) determinabitur. </s>
  <s xml:id="echoid-s7440" xml:space="preserve">rem appoſita figura <lb/>ſatìs exprimit.</s>
  <s xml:id="echoid-s7441" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7442" xml:space="preserve">Hæc autem omnia de ſuprà comprobatis dilucidè conſectantur.</s>
  <s xml:id="echoid-s7443" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7444" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s7445" xml:space="preserve">Ac ità quidem circa ſimplices planas ſuperficies refringentes <lb/>
<anchor type="note" xlink:label="note-0131-01a" xlink:href="note-0131-01"/>
ſeſe res habet. </s>
  <s xml:id="echoid-s7446" xml:space="preserve">Quòd ſi corpori parallelis planis MN, μν terminato <lb/>exponatur recta FG; </s>
  <s xml:id="echoid-s7447" xml:space="preserve">Sint rectæ FP, GQ, ADBO ipſi PQ <lb/>perpendiculares, &amp; </s>
  <s xml:id="echoid-s7448" xml:space="preserve">fiat BD. </s>
  <s xml:id="echoid-s7449" xml:space="preserve">BS :</s>
  <s xml:id="echoid-s7450" xml:space="preserve">: I.</s>
  <s xml:id="echoid-s7451" xml:space="preserve">R; </s>
  <s xml:id="echoid-s7452" xml:space="preserve">adſumatúrque Aα = DS; <lb/></s>
  <s xml:id="echoid-s7453" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7454" xml:space="preserve">fiat AB. </s>
  <s xml:id="echoid-s7455" xml:space="preserve">αβ :</s>
  <s xml:id="echoid-s7456" xml:space="preserve">: FP. </s>
  <s xml:id="echoid-s7457" xml:space="preserve">XP; </s>
  <s xml:id="echoid-s7458" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7459" xml:space="preserve">per X, α ducatur recta X α Y, erit <lb/>X α Y lineæ FAG imago abſoluta. </s>
  <s xml:id="echoid-s7460" xml:space="preserve">Ergò ejus imago ad oculum O <lb/>relata (in hoc caſu) citra ipſam X α Y verſus ſuperficiem μν nonnihil <lb/>incurvata diſponetur, qualem exhibet linea φαγ. </s>
  <s xml:id="echoid-s7461" xml:space="preserve">id quod ex eo ſatìs <lb/>videtur liquere, quòd recta X α Y ſit imago reſpectu oculiin ipſa <lb/>OB à B infinitè ſemoti. </s>
  <s xml:id="echoid-s7462" xml:space="preserve">deſignari verò poterit hæc imago ad hunc <lb/>modum. </s>
  <s xml:id="echoid-s7463" xml:space="preserve">ſit fag (minuſculis elementis indigitata) imago rectæ FAG <lb/>ad ſuperſiciem refringentem μν relata (hoc eſt ad oculos in refractis <lb/>f μ M, q ν N, a DB, reliquíſque, nec non in medio μν MN verſus <lb/>O protenſo, ſitos) juxta proximè commonſtrata delineabilis. </s>
  <s xml:id="echoid-s7464" xml:space="preserve">tum <lb/>hujus ipſius _fag_ velut in medio MN μν verſus A protenſo poſitæ, ex <lb/>refractione ad ſuperficiem MN emergens, &amp; </s>
  <s xml:id="echoid-s7465" xml:space="preserve">ad oculum O relata <lb/>conſtruatur imago φαγ (itidem ad modum nuperrimè præſcriptum) <lb/>hæc rectam FAG per corpus MN μν ſpectatam repræſentabit. </s>
  <s xml:id="echoid-s7466" xml:space="preserve">ex-<lb/>perientia teſtis advocetur, ego pluribus in re perplexiore, quàm uti-<lb/>liore ſuperſedeo.</s>
  <s xml:id="echoid-s7467" xml:space="preserve"/>
</p>
<div xml:id="echoid-div193" type="float" level="2" n="5">
<note position="right" xlink:label="note-0131-01" xlink:href="note-0131-01a" xml:space="preserve">Fig. 185.</note>
</div>
<p>
  <s xml:id="echoid-s7468" xml:space="preserve">X. </s>
  <s xml:id="echoid-s7469" xml:space="preserve">Porrò quod _plana ſpecula_ (ſimplicia, vel compoſita) attinet, <lb/>in iis palàm eſt imagines abſolutas ac relatas omnino ſibi coincidere; <lb/></s>
  <s xml:id="echoid-s7470" xml:space="preserve">quo fit, ut eæ objectorum magnitudines, figuras, diſtantias (ſitu
<pb o="114" file="0132" n="132" rhead=""/>
tamen nonnunquam inverſo) quàm exactiſſimè referant. </s>
  <s xml:id="echoid-s7471" xml:space="preserve">qua de re <lb/>(tam facili, toties acta) penitùs reticens ad minùs trita me pro-<lb/>moveo.</s>
  <s xml:id="echoid-s7472" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7473" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s7474" xml:space="preserve">Sit jam _Circulare Speculum convexum_ DMB, cujus centrum C; <lb/></s>
  <s xml:id="echoid-s7475" xml:space="preserve">
<anchor type="note" xlink:label="note-0132-01a" xlink:href="note-0132-01"/>
&amp; </s>
  <s xml:id="echoid-s7476" xml:space="preserve">per C protendatur recta CBA; </s>
  <s xml:id="echoid-s7477" xml:space="preserve">in qua ſumatur portio quædam AR, <lb/>fiátque CA. </s>
  <s xml:id="echoid-s7478" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s7479" xml:space="preserve">: CX. </s>
  <s xml:id="echoid-s7480" xml:space="preserve">XB; </s>
  <s xml:id="echoid-s7481" xml:space="preserve">neq; </s>
  <s xml:id="echoid-s7482" xml:space="preserve">non CR. </s>
  <s xml:id="echoid-s7483" xml:space="preserve">RB :</s>
  <s xml:id="echoid-s7484" xml:space="preserve">: CY. </s>
  <s xml:id="echoid-s7485" xml:space="preserve">YB; </s>
  <s xml:id="echoid-s7486" xml:space="preserve">erit YX imago <lb/>abſoluta rectæ RA; </s>
  <s xml:id="echoid-s7487" xml:space="preserve">quòd ſi CB biſecetur in Z; </s>
  <s xml:id="echoid-s7488" xml:space="preserve">erit BZ totius BA ad <lb/>infinitum exporrectæ imago abſoluta; </s>
  <s xml:id="echoid-s7489" xml:space="preserve">hoc eſt, illæ tales erunt oculi <lb/>reſpectu in ipſa AB conſtituti. </s>
  <s xml:id="echoid-s7490" xml:space="preserve">ſecùs autem uſpiam collocato oculo, <lb/>tanquam ad O, totius AB quod conſpicuum eſt (hoc eſt quod ſupra <lb/>horizontem OT, ſpeculo contiguum extat) ſupra citráque XB ap-<lb/>parebit. </s>
  <s xml:id="echoid-s7491" xml:space="preserve">Enimvero tranſeat radii AM reflexus KMO per O; </s>
  <s xml:id="echoid-s7492" xml:space="preserve">ita-<lb/>que punctum K (quod olim oſtenſum) ſupra punctum X, verſus A, <lb/>extat. </s>
  <s xml:id="echoid-s7493" xml:space="preserve">quinetiam (ex indidem monftratis) puncti A imagines omnes, <lb/>oculum O reſpicientes, ex reflexione factæ ad partes BMD, citra <lb/>CA verſus O, cadunt. </s>
  <s xml:id="echoid-s7494" xml:space="preserve">ejus igitur imago quæ in OK, puta α, in ipſa <lb/>KM exiſtet (id quod etiam, nè quis dubitet, exertius mox oſten-<lb/>demus). </s>
  <s xml:id="echoid-s7495" xml:space="preserve">ſimili ratione puncti R imago, cogitaρ, ſupra Y, citráque <lb/>BY jacet. </s>
  <s xml:id="echoid-s7496" xml:space="preserve">quòd ſi porrò per O tranſeat recta ODLH, quæ reflexa ſit <lb/>rectæ DS ad CA parallelæ (hæc autem quomodo ducatur, antehac <lb/>declaratum habetur) erit in ODL imago puncti (quale concipiatur <lb/>S) in ipſa AB infinitè ſemoti; </s>
  <s xml:id="echoid-s7497" xml:space="preserve">hæc puta ſit ad σ. </s>
  <s xml:id="echoid-s7498" xml:space="preserve">erit itaque curva <lb/>B ρασ imago totius infinitæ rectæ BAS, ad oculum O relata.</s>
  <s xml:id="echoid-s7499" xml:space="preserve"/>
</p>
<div xml:id="echoid-div194" type="float" level="2" n="6">
<note position="left" xlink:label="note-0132-01" xlink:href="note-0132-01a" xml:space="preserve">Fig. 186.</note>
</div>
<p>
  <s xml:id="echoid-s7500" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s7501" xml:space="preserve">Iſta verò linea tali pacto delineatur: </s>
  <s xml:id="echoid-s7502" xml:space="preserve">Super diametrum CO <lb/>deſcribatur circulus OTC; </s>
  <s xml:id="echoid-s7503" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7504" xml:space="preserve">ab O ducatur recta quæpiam OMF, <lb/>cujus reflexa ſit MA; </s>
  <s xml:id="echoid-s7505" xml:space="preserve">in qua ſumatur ME = MF; </s>
  <s xml:id="echoid-s7506" xml:space="preserve">tum ſecetur <lb/>FM in α, ut ſit F α. </s>
  <s xml:id="echoid-s7507" xml:space="preserve">α M :</s>
  <s xml:id="echoid-s7508" xml:space="preserve">: AE. </s>
  <s xml:id="echoid-s7509" xml:space="preserve">AM; </s>
  <s xml:id="echoid-s7510" xml:space="preserve">erit (è pridem demonſtra-<lb/>tis) punctum α puncti A imago. </s>
  <s xml:id="echoid-s7511" xml:space="preserve">ſimili modo quotcunque lineæ <lb/>B αρσ puncta reperiuntur.</s>
  <s xml:id="echoid-s7512" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7513" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s7514" xml:space="preserve">Quòd autem ſit punctum α citra K (verſus oculum) ità con-<lb/>ſtabit. </s>
  <s xml:id="echoid-s7515" xml:space="preserve">Ducatur FQ ad AM parallela. </s>
  <s xml:id="echoid-s7516" xml:space="preserve">eſt ergo angulus FQA <lb/>par angulo CAM. </s>
  <s xml:id="echoid-s7517" xml:space="preserve">aſt angulus FCA angulo ACE minor eſt. </s>
  <s xml:id="echoid-s7518" xml:space="preserve">ergò <lb/>eſt CF. </s>
  <s xml:id="echoid-s7519" xml:space="preserve">FQ &amp;</s>
  <s xml:id="echoid-s7520" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7521" xml:space="preserve">CE. </s>
  <s xml:id="echoid-s7522" xml:space="preserve">AE. </s>
  <s xml:id="echoid-s7523" xml:space="preserve">atqui CF = CE; </s>
  <s xml:id="echoid-s7524" xml:space="preserve">quare FQ &amp;</s>
  <s xml:id="echoid-s7525" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s7526" xml:space="preserve">AE. <lb/></s>
  <s xml:id="echoid-s7527" xml:space="preserve">ergò eſt FQ. </s>
  <s xml:id="echoid-s7528" xml:space="preserve">AM &amp;</s>
  <s xml:id="echoid-s7529" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s7530" xml:space="preserve">AE. </s>
  <s xml:id="echoid-s7531" xml:space="preserve">AM. </s>
  <s xml:id="echoid-s7532" xml:space="preserve">hoc eſt FK. </s>
  <s xml:id="echoid-s7533" xml:space="preserve">KM &amp;</s>
  <s xml:id="echoid-s7534" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s7535" xml:space="preserve">F α. </s>
  <s xml:id="echoid-s7536" xml:space="preserve">αM. </s>
  <s xml:id="echoid-s7537" xml:space="preserve"><lb/>componendóque FM. </s>
  <s xml:id="echoid-s7538" xml:space="preserve">KM &amp;</s>
  <s xml:id="echoid-s7539" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s7540" xml:space="preserve">FM. </s>
  <s xml:id="echoid-s7541" xml:space="preserve">αM. </s>
  <s xml:id="echoid-s7542" xml:space="preserve">unde KM &amp;</s>
  <s xml:id="echoid-s7543" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7544" xml:space="preserve">αM. </s>
  <s xml:id="echoid-s7545" xml:space="preserve">adeó-<lb/>que punctum αcitra K verſus O jacet: </s>
  <s xml:id="echoid-s7546" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s7547" xml:space="preserve">E. </s>
  <s xml:id="echoid-s7548" xml:space="preserve">D.</s>
  <s xml:id="echoid-s7549" xml:space="preserve"/>
</p>
<pb o="115" file="0133" n="133" rhead=""/>
<p>
  <s xml:id="echoid-s7550" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s7551" xml:space="preserve">Exhinc _Euclidis, Alhazeni_, communíſque fermè ſententia <lb/>convellitur, quæ rectæ BA rectam BK, infinitæque BS ipſam BL <lb/>imagines ſtatuit; </s>
  <s xml:id="echoid-s7552" xml:space="preserve">proindéque corruunt omnia, quæ principio ſuper-<lb/>extruunt iſti gratìs adſumpto, rationíque diſſentaneo. </s>
  <s xml:id="echoid-s7553" xml:space="preserve">Veruntamen <lb/>_Opticorum noviſſimus ſcriptor, eruditiſſimúſque vir_, veterum ipſe veſti-<lb/>giis inſiſtens poſtulatum iſtud ab experientia ſtabilitum vult, ejúſque <lb/>veritatem ſeſe deprædicat centies explorâſſe; </s>
  <s xml:id="echoid-s7554" xml:space="preserve">doctrinam itaque no-<lb/>ſtram invicto ſensûs teſtimonio refutavit. </s>
  <s xml:id="echoid-s7555" xml:space="preserve">atqui repono, non potuiſſe <lb/>illum quantumvis oculatum &amp; </s>
  <s xml:id="echoid-s7556" xml:space="preserve">ſagacem quod obtendit vel ſemel ex-<lb/>plorare. </s>
  <s xml:id="echoid-s7557" xml:space="preserve">nec hoc in caſu poterit doctrina noſtra tentari, nedum re-<lb/>felli. </s>
  <s xml:id="echoid-s7558" xml:space="preserve">nam (præterquam quòd perpendicularis CBA ſitum exactè <lb/>dignoſcere perquam arduum, forſan impoſſibile fuerit) quum lineola <lb/>B αρσ infinitam, juxta nos, lineam rectam BS repræſentet, ipſúm-<lb/>que punctum σ (infinitè diſſito puncto S reſpondens, atque rectam <lb/>DH biſecans) à puncto L modicè diſtet, quæ amabò visûs acies <lb/>curvæ B ασ à recta BL deflectionem cernat? </s>
  <s xml:id="echoid-s7559" xml:space="preserve">itaque fruſtrà eſſe vide-<lb/>tur acutiſſimus vir, ad teſtem provocans hac in parte minùs com-<lb/>petentem, déque cujus ſententia vix ullatenus conſtare poſſit. </s>
  <s xml:id="echoid-s7560" xml:space="preserve">Sanè <lb/>quoad affinem in _Dioptricis_ caſum, quem attigimus ſupra, demiſſo in <lb/>aquam perpendiculo, oculo ſimpliciter inſpectanti, videbitur ejus ima-<lb/>go nihil quicquam à perpendiculari declinans; </s>
  <s xml:id="echoid-s7561" xml:space="preserve">verùm ope reflectionis <lb/>juſtum perpendicularis ſitum obſervando (qui nudo ſcilicet obtutu <lb/>planè dijudicari nequit) notoriè deprehenditur aquæ immerſi perpen-<lb/>diculi imago ab ipſo deviare. </s>
  <s xml:id="echoid-s7562" xml:space="preserve">neque dubito quin pariter in præſente <lb/>caſu ritè conſulta experientia pro nobis ſit pronunciatura. </s>
  <s xml:id="echoid-s7563" xml:space="preserve">Quinimò <lb/>noſtris ex effatis (luculentâ opinor ratione ſuffultis) apparebit, unde <lb/>principium illud multis in caſibus experientiæ videatur conſentire; <lb/></s>
  <s xml:id="echoid-s7564" xml:space="preserve">quoniam nempe contingit, ut in iis à vero non multùm abſcedat; </s>
  <s xml:id="echoid-s7565" xml:space="preserve"><lb/>ejúſque proinde falſitatem ſenſus (niſi ratione, vel certiore ſenſu adju-<lb/>tus) perſpicere nequeat. </s>
  <s xml:id="echoid-s7566" xml:space="preserve">aſt exorbito.</s>
  <s xml:id="echoid-s7567" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7568" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s7569" xml:space="preserve">Sit rurſus _Speculum concavum_ BMD; </s>
  <s xml:id="echoid-s7570" xml:space="preserve">cujus centrum C, &amp; </s>
  <s xml:id="echoid-s7571" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0133-01a" xlink:href="note-0133-01"/>
per C extendatur infinita recta CBL, biſecetúrque ſemidiameter <lb/>CB in Z; </s>
  <s xml:id="echoid-s7572" xml:space="preserve">ac in ZB ſumptis quibuſcunque punctis A, R; </s>
  <s xml:id="echoid-s7573" xml:space="preserve">fiat CA. <lb/></s>
  <s xml:id="echoid-s7574" xml:space="preserve">AB :</s>
  <s xml:id="echoid-s7575" xml:space="preserve">: CX. </s>
  <s xml:id="echoid-s7576" xml:space="preserve">XB; </s>
  <s xml:id="echoid-s7577" xml:space="preserve">itémque CR. </s>
  <s xml:id="echoid-s7578" xml:space="preserve">RB :</s>
  <s xml:id="echoid-s7579" xml:space="preserve">: CY. </s>
  <s xml:id="echoid-s7580" xml:space="preserve">YB; </s>
  <s xml:id="echoid-s7581" xml:space="preserve">erit quideminfi-<lb/>nita BL totius BZ imago abſoluta, &amp; </s>
  <s xml:id="echoid-s7582" xml:space="preserve">portio YX portionis RA; </s>
  <s xml:id="echoid-s7583" xml:space="preserve"><lb/>verùm extra axem BC uſpiam conſtituto viſu, velut ad O, ad hunc <lb/>relatæ ipſius ZB, ejúſque partium imagines ità determinantur.</s>
  <s xml:id="echoid-s7584" xml:space="preserve"/>
</p>
<div xml:id="echoid-div195" type="float" level="2" n="7">
<note position="right" xlink:label="note-0133-01" xlink:href="note-0133-01a" xml:space="preserve">Fig. 187.</note>
</div>
<pb o="116" file="0134" n="134" rhead=""/>
<p>
  <s xml:id="echoid-s7585" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s7586" xml:space="preserve">Addiametrum CO deſcribatur circulus CFH; </s>
  <s xml:id="echoid-s7587" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7588" xml:space="preserve">ab O <lb/>
<anchor type="note" xlink:label="note-0134-01a" xlink:href="note-0134-01"/>
radiusincidat talis, ut cum ejus reflexus ſit DS, contingat fore DS <lb/> = {1/2} DH, vel {1/2} DI; </s>
  <s xml:id="echoid-s7589" xml:space="preserve">poſitâ CI ad DS perpendiculari (talis autem <lb/>radius facilè duci poſſe concipiatur; </s>
  <s xml:id="echoid-s7590" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7591" xml:space="preserve">per curvam appropriatam re-<lb/>verà ſtatim determinetur; </s>
  <s xml:id="echoid-s7592" xml:space="preserve">id proinde nos non diſtinebit). </s>
  <s xml:id="echoid-s7593" xml:space="preserve">Erit tum <lb/>puncti Simago, puta σ, à puncto D infinitè disjuncta; </s>
  <s xml:id="echoid-s7594" xml:space="preserve">quoniam (id <lb/>quod fieri nequit, niſi H σ, σ D ſint infinitæ) eſt H σ. </s>
  <s xml:id="echoid-s7595" xml:space="preserve">σ D :</s>
  <s xml:id="echoid-s7596" xml:space="preserve">: IS. <lb/></s>
  <s xml:id="echoid-s7597" xml:space="preserve">SD. </s>
  <s xml:id="echoid-s7598" xml:space="preserve">Jam in arcum DB cadat utcunque radius OM, cujus reflexus <lb/>ſit MA E; </s>
  <s xml:id="echoid-s7599" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7600" xml:space="preserve">in hac ſumatur ME = MF; </s>
  <s xml:id="echoid-s7601" xml:space="preserve">tum in OM producta <lb/>capiatur punctum α, ut ſit F α. </s>
  <s xml:id="echoid-s7602" xml:space="preserve">α M :</s>
  <s xml:id="echoid-s7603" xml:space="preserve">: EA. </s>
  <s xml:id="echoid-s7604" xml:space="preserve">AM; </s>
  <s xml:id="echoid-s7605" xml:space="preserve">erit α puncti A <lb/>imago. </s>
  <s xml:id="echoid-s7606" xml:space="preserve">ſimili methodo reperiatur ρ puncti R imago; </s>
  <s xml:id="echoid-s7607" xml:space="preserve">neque non reliqua <lb/>totius R ρασ, ipſam BS referentis, puncta.</s>
  <s xml:id="echoid-s7608" xml:space="preserve"/>
</p>
<div xml:id="echoid-div196" type="float" level="2" n="8">
<note position="left" xlink:label="note-0134-01" xlink:href="note-0134-01a" xml:space="preserve">Fig. 187.</note>
</div>
<p>
  <s xml:id="echoid-s7609" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s7610" xml:space="preserve">In harc verò conſtructionem quædam veniunt adnotanda.</s>
  <s xml:id="echoid-s7611" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Fig. 187, <lb/>188.</note>
<p>
  <s xml:id="echoid-s7612" xml:space="preserve">1. </s>
  <s xml:id="echoid-s7613" xml:space="preserve">Quòd CS &amp;</s>
  <s xml:id="echoid-s7614" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7615" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s7616" xml:space="preserve">Nam 4 CZq = CBq = 3 SDq + <lb/>CSq. </s>
  <s xml:id="echoid-s7617" xml:space="preserve">ergò quum ſit CZ &amp;</s>
  <s xml:id="echoid-s7618" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7619" xml:space="preserve">SD; </s>
  <s xml:id="echoid-s7620" xml:space="preserve">erit CS &amp;</s>
  <s xml:id="echoid-s7621" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7622" xml:space="preserve">CZ.</s>
  <s xml:id="echoid-s7623" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7624" xml:space="preserve">2. </s>
  <s xml:id="echoid-s7625" xml:space="preserve">Quod CA &amp;</s>
  <s xml:id="echoid-s7626" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7627" xml:space="preserve">CS. </s>
  <s xml:id="echoid-s7628" xml:space="preserve">Nam (è ſuprà monſtratis) ſi ducatur <lb/>recta M ψ ad DO parallela, ejuſce reflexa (puta M ξ) ſecabit ipſam <lb/>DS, verſus I, puta ad ξ. </s>
  <s xml:id="echoid-s7629" xml:space="preserve">ergò M ξ ipſam CB ſecabit ſupra punctum S, <lb/>velut ad φ. </s>
  <s xml:id="echoid-s7630" xml:space="preserve">atqui quoniam ang. </s>
  <s xml:id="echoid-s7631" xml:space="preserve">CMO &amp;</s>
  <s xml:id="echoid-s7632" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7633" xml:space="preserve">CMψ, ſeu ang. </s>
  <s xml:id="echoid-s7634" xml:space="preserve">CMA <lb/>&amp;</s>
  <s xml:id="echoid-s7635" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7636" xml:space="preserve">CMφ, eſt CA &amp;</s>
  <s xml:id="echoid-s7637" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7638" xml:space="preserve">C φ; </s>
  <s xml:id="echoid-s7639" xml:space="preserve">adeóque magìs eſt CA &amp;</s>
  <s xml:id="echoid-s7640" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7641" xml:space="preserve">CS.</s>
  <s xml:id="echoid-s7642" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7643" xml:space="preserve">3. </s>
  <s xml:id="echoid-s7644" xml:space="preserve">Quòd EA &amp;</s>
  <s xml:id="echoid-s7645" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7646" xml:space="preserve">AM. </s>
  <s xml:id="echoid-s7647" xml:space="preserve">cùm enim ſit EM (vel FM) &amp;</s>
  <s xml:id="echoid-s7648" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7649" xml:space="preserve">HD, <lb/>atque DS &amp;</s>
  <s xml:id="echoid-s7650" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7651" xml:space="preserve">MA; </s>
  <s xml:id="echoid-s7652" xml:space="preserve">erit EM. </s>
  <s xml:id="echoid-s7653" xml:space="preserve">MA &amp;</s>
  <s xml:id="echoid-s7654" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7655" xml:space="preserve">FD. </s>
  <s xml:id="echoid-s7656" xml:space="preserve">DS :</s>
  <s xml:id="echoid-s7657" xml:space="preserve">: 2. </s>
  <s xml:id="echoid-s7658" xml:space="preserve">1.</s>
  <s xml:id="echoid-s7659" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7660" xml:space="preserve">4. </s>
  <s xml:id="echoid-s7661" xml:space="preserve">Hinc denuò liquebit totam lineam B ρασ ultra rectam CBL <lb/>jacere. </s>
  <s xml:id="echoid-s7662" xml:space="preserve">nam ducatur FQ ad AM parallela eſt hîc ang FCA &amp;</s>
  <s xml:id="echoid-s7663" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7664" xml:space="preserve">ang <lb/>ACE. </s>
  <s xml:id="echoid-s7665" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7666" xml:space="preserve">ang FQA = ang CAE. </s>
  <s xml:id="echoid-s7667" xml:space="preserve">quapropter erit CF. </s>
  <s xml:id="echoid-s7668" xml:space="preserve">FQ <lb/>&amp;</s>
  <s xml:id="echoid-s7669" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s7670" xml:space="preserve">CE. </s>
  <s xml:id="echoid-s7671" xml:space="preserve">AE. </s>
  <s xml:id="echoid-s7672" xml:space="preserve">adeóque FQ &amp;</s>
  <s xml:id="echoid-s7673" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7674" xml:space="preserve">AE. </s>
  <s xml:id="echoid-s7675" xml:space="preserve">acindè FQ. </s>
  <s xml:id="echoid-s7676" xml:space="preserve">AM &amp;</s>
  <s xml:id="echoid-s7677" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7678" xml:space="preserve">AE. <lb/></s>
  <s xml:id="echoid-s7679" xml:space="preserve">AM; </s>
  <s xml:id="echoid-s7680" xml:space="preserve">hoc eſt FK, KM &amp;</s>
  <s xml:id="echoid-s7681" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7682" xml:space="preserve">Fα. </s>
  <s xml:id="echoid-s7683" xml:space="preserve">αM. </s>
  <s xml:id="echoid-s7684" xml:space="preserve">dividendóque FM. </s>
  <s xml:id="echoid-s7685" xml:space="preserve">KM &amp;</s>
  <s xml:id="echoid-s7686" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7687" xml:space="preserve"><lb/>FM. </s>
  <s xml:id="echoid-s7688" xml:space="preserve">αM. </s>
  <s xml:id="echoid-s7689" xml:space="preserve">quare α M &amp;</s>
  <s xml:id="echoid-s7690" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7691" xml:space="preserve">KM. </s>
  <s xml:id="echoid-s7692" xml:space="preserve">adeóque punctum α ultra K in <lb/>recta OK protenſa jacet.</s>
  <s xml:id="echoid-s7693" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7694" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s7695" xml:space="preserve">Quòd ſiad partes alteras rectæ OD ducatur radius ON’ <lb/>cujus reflexus NGT = NV; </s>
  <s xml:id="echoid-s7696" xml:space="preserve">ſitque TG. </s>
  <s xml:id="echoid-s7697" xml:space="preserve">GN &amp;</s>
  <s xml:id="echoid-s7698" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s7699" xml:space="preserve">2. </s>
  <s xml:id="echoid-s7700" xml:space="preserve">1; </s>
  <s xml:id="echoid-s7701" xml:space="preserve">ſtatuen-<lb/>da eſt puncti Gimago (puta γ) ad partes O. </s>
  <s xml:id="echoid-s7702" xml:space="preserve">quinimò cùm in hanc <lb/>rem plura ſubjici poſſent, ego jam _Specimina_ tantùm inſtituens <lb/>(quippe cùm operâ dignum haud arbitrer adeò tenuem materiam curi-<lb/>oſiùs proſequi) à minutiis abſtineo. </s>
  <s xml:id="echoid-s7703" xml:space="preserve">quo &amp; </s>
  <s xml:id="echoid-s7704" xml:space="preserve">indè pronior ſum, quo-<lb/>niam in hac re copioſus videtur A. </s>
  <s xml:id="echoid-s7705" xml:space="preserve">_Tacquetus;_ </s>
  <s xml:id="echoid-s7706" xml:space="preserve">ſubinde quidem is, <lb/>ob admiſſum iſtud falſum principium, ceſpitans, at bene multa credo
<pb o="117" file="0135" n="135" rhead=""/>
ſuggerens haud aſpernanda. </s>
  <s xml:id="echoid-s7707" xml:space="preserve">relinquantur igitur ei cætera, mihi <lb/>ſuffecerit, quòd veriorem _phænomena{εμ}<unsure/>_ detegendi declarandíque me-<lb/>thodum adniſus ſim aliquatenus enucleare. </s>
  <s xml:id="echoid-s7708" xml:space="preserve">pergamus ad alios caſus, <lb/>haud ità pertractatos.</s>
  <s xml:id="echoid-s7709" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7710" xml:space="preserve">XIX Objiciatur ſpeculo MBND recta FAG, rectæ CA (per <lb/>ſpeculi centrum C tranſeunti) perpendicularis; </s>
  <s xml:id="echoid-s7711" xml:space="preserve">adverto, ſi fuerit ipſa <lb/>
<anchor type="note" xlink:label="note-0135-01a" xlink:href="note-0135-01"/>
CA major quàm CZ, quadrans diametri BD, quòd rectæ FAG <lb/>ad infinitum utrinque protractæ ad totum circulum (ejus ad partes <lb/>intelligo concavas ſimul acconvexas) imago abſoluta (quinetiam ima-<lb/>go ad oculum in ipſo centro C conſtitutum relata) erit _Ellipſis_. </s>
  <s xml:id="echoid-s7712" xml:space="preserve">item <lb/>ſi CA minor ſit, quàm CZ, quòd ipſius FA G imago abſoluta <lb/>(vel dicto modo relata) conſtabit ex hyperbolis oppoſitis; </s>
  <s xml:id="echoid-s7713" xml:space="preserve">ſi denuò <lb/>CA ipſam CZ adæquet (vel FG per ipſum Z tranſeat) quòd ad <lb/>parabolam ejuſmodi conſiſtet imago. </s>
  <s xml:id="echoid-s7714" xml:space="preserve">Sed modum tranſgrederer hæc <lb/>jam aggrediens demonſtrare. </s>
  <s xml:id="echoid-s7715" xml:space="preserve">Expectent igitur.</s>
  <s xml:id="echoid-s7716" xml:space="preserve">‖</s>
</p>
<div xml:id="echoid-div197" type="float" level="2" n="9">
<note position="right" xlink:label="note-0135-01" xlink:href="note-0135-01a" xml:space="preserve">Fig. 189.</note>
</div>
</div>
<div xml:id="echoid-div199" type="section" level="1" n="24">
<head xml:id="echoid-head27" xml:space="preserve"><emph style="sc">Lect.</emph> XVII.</head>
<p>
  <s xml:id="echoid-s7717" xml:space="preserve">1. </s>
  <s xml:id="echoid-s7718" xml:space="preserve">ADea, quæ ſub finitam præcedentem propoſuimus demon-<lb/>ſtranda _neceſſariam, alioquin notabilem, Conicarum Sectionum_ <lb/>_proprietatem_ imprimìs oſtendemus.</s>
  <s xml:id="echoid-s7719" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7720" xml:space="preserve">Sit triangulum ACE, rectum habens angulum ad C; </s>
  <s xml:id="echoid-s7721" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7722" xml:space="preserve">inde-<lb/>
<anchor type="note" xlink:label="note-0135-02a" xlink:href="note-0135-02"/>
finitè protractis lateribus AC, AE, in AC ſumatur quod piam pun-<lb/>ctum X, ducatúrque XG ad CE parallela; </s>
  <s xml:id="echoid-s7723" xml:space="preserve">inſeratur autem angulo <lb/>CXG recta CZ æqualis ipſi XG; </s>
  <s xml:id="echoid-s7724" xml:space="preserve">dico punctum indeterminatum Z ad <lb/>ſectionum conicarum aliquam conſiſtere.</s>
  <s xml:id="echoid-s7725" xml:space="preserve"/>
</p>
<div xml:id="echoid-div199" type="float" level="2" n="1">
<note position="right" xlink:label="note-0135-02" xlink:href="note-0135-02a" xml:space="preserve">Fig. 190.</note>
</div>
<p>
  <s xml:id="echoid-s7726" xml:space="preserve">II. </s>
  <s xml:id="echoid-s7727" xml:space="preserve">Nempe primò, ſit angulus A ſemirecto minor (vel AC &amp;</s>
  <s xml:id="echoid-s7728" xml:space="preserve">gt; <lb/></s>
  <s xml:id="echoid-s7729" xml:space="preserve">CE) erit punctum Z ad ellipſin, quæ determinatur hoc pacto: </s>
  <s xml:id="echoid-s7730" xml:space="preserve">An-<lb/>guli LCP ſemirecti fiant (ad utramque rectæ CE partem) liquet <lb/>igitur rectas CP ipſi AE occurrere, puta ad puncta R, &amp; </s>
  <s xml:id="echoid-s7731" xml:space="preserve">S. </s>
  <s xml:id="echoid-s7732" xml:space="preserve">ab his
<pb o="118" file="0136" n="136" rhead=""/>
ad ipſam EC parallelæ ducantur rectæ RT, SV; </s>
  <s xml:id="echoid-s7733" xml:space="preserve">palàm eſt indeter-<lb/>minatum punctum X inter limites T, V conſiſtere (nam extra TV <lb/>punctum quodlibet L accipiendo, &amp; </s>
  <s xml:id="echoid-s7734" xml:space="preserve">indè ducendo LI P ad CE paralle-<lb/>lam, erit CL, hoc eft LP, major quàm LI, unde à C ad rectam LI, <lb/>nulla duci recta poteſt æqualis ipſi LI). </s>
  <s xml:id="echoid-s7735" xml:space="preserve">Jam autem dico, quòd <lb/>punctum Z ad ellipſin exiſtit, cujus axis TV, focus C. </s>
  <s xml:id="echoid-s7736" xml:space="preserve">Nam biſe-<lb/>cetur TV in K; </s>
  <s xml:id="echoid-s7737" xml:space="preserve">fiat VD = TC; </s>
  <s xml:id="echoid-s7738" xml:space="preserve">ducatur KH ad CE parallela; <lb/></s>
  <s xml:id="echoid-s7739" xml:space="preserve">per H ducatur HN ad CK parallela. </s>
  <s xml:id="echoid-s7740" xml:space="preserve">Eſtque KH = {TR + VS/2} = <lb/>{CT + CV/2} = KT = KV. </s>
  <s xml:id="echoid-s7741" xml:space="preserve">Et quoniam AV. </s>
  <s xml:id="echoid-s7742" xml:space="preserve">AT :</s>
  <s xml:id="echoid-s7743" xml:space="preserve">: (VS. </s>
  <s xml:id="echoid-s7744" xml:space="preserve"><lb/>TR (hoc eſt) :</s>
  <s xml:id="echoid-s7745" xml:space="preserve">: CV. </s>
  <s xml:id="echoid-s7746" xml:space="preserve">CT :</s>
  <s xml:id="echoid-s7747" xml:space="preserve">:) CV. </s>
  <s xml:id="echoid-s7748" xml:space="preserve">DV; </s>
  <s xml:id="echoid-s7749" xml:space="preserve">erit per rationis con-<lb/>vcrſionem AV. </s>
  <s xml:id="echoid-s7750" xml:space="preserve">TV :</s>
  <s xml:id="echoid-s7751" xml:space="preserve">: CV. </s>
  <s xml:id="echoid-s7752" xml:space="preserve">CD. </s>
  <s xml:id="echoid-s7753" xml:space="preserve">vel, conſequentes ſubduplando, <lb/>AV. </s>
  <s xml:id="echoid-s7754" xml:space="preserve">KV :</s>
  <s xml:id="echoid-s7755" xml:space="preserve">: CV. </s>
  <s xml:id="echoid-s7756" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s7757" xml:space="preserve">dividendóque AK. </s>
  <s xml:id="echoid-s7758" xml:space="preserve">KV :</s>
  <s xml:id="echoid-s7759" xml:space="preserve">: KV. </s>
  <s xml:id="echoid-s7760" xml:space="preserve">CK; </s>
  <s xml:id="echoid-s7761" xml:space="preserve">hoc eſt <lb/>AK. </s>
  <s xml:id="echoid-s7762" xml:space="preserve">KH :</s>
  <s xml:id="echoid-s7763" xml:space="preserve">: KH. </s>
  <s xml:id="echoid-s7764" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s7765" xml:space="preserve">hoc eſt HN. </s>
  <s xml:id="echoid-s7766" xml:space="preserve">NG :</s>
  <s xml:id="echoid-s7767" xml:space="preserve">: KH. </s>
  <s xml:id="echoid-s7768" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s7769" xml:space="preserve">quare <lb/>KH x NG = CK x HN = CK x KX. </s>
  <s xml:id="echoid-s7770" xml:space="preserve">atqui eſt CZq = XGq <lb/> = KHq + NGq + 2 KH x NG. </s>
  <s xml:id="echoid-s7771" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7772" xml:space="preserve">CXq = CKq + KXq <lb/>+ 2 CK x KX = CKq + KXq + 2 KH x NG. </s>
  <s xml:id="echoid-s7773" xml:space="preserve">ergo <lb/>KHq + NGq - CKq - KXq = CZq - CXq = XZq. </s>
  <s xml:id="echoid-s7774" xml:space="preserve"><lb/>Ad alteras biſegmenti K partes ſumatur K ξ = KX, ducatúrque ξν ad <lb/>KH parallela, ſecans curvam TEZV in ζ, &amp; </s>
  <s xml:id="echoid-s7775" xml:space="preserve">rectam AH in γ, ac <lb/>ipſam NH in ν erit quoque, ſimili ex diſcurſu, ξζq = KHq + <lb/>νγq - CKq - Kξq; </s>
  <s xml:id="echoid-s7776" xml:space="preserve">unde liquet fore ξζ = XZ; </s>
  <s xml:id="echoid-s7777" xml:space="preserve">connexíſque <lb/>proinde rectis Cζ, Dζ, erit Dζ = CZ; </s>
  <s xml:id="echoid-s7778" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7779" xml:space="preserve">Cζ + CZ = ξγ + <lb/>XG = 2 KH = TV. </s>
  <s xml:id="echoid-s7780" xml:space="preserve">ergò Cζ + Dζ (vel DZ + CZ) = TV. </s>
  <s xml:id="echoid-s7781" xml:space="preserve"><lb/>unde perſpicitur _curvam TζZV eſſe ellipſin_, cujus _axis_ TV; </s>
  <s xml:id="echoid-s7782" xml:space="preserve">_foci_ <lb/>C, D.</s>
  <s xml:id="echoid-s7783" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7784" xml:space="preserve">III. </s>
  <s xml:id="echoid-s7785" xml:space="preserve">Sit autem ſecundò angulus CA E major ſemirecto (vel AC <lb/>
<anchor type="note" xlink:label="note-0136-01a" xlink:href="note-0136-01"/>
&amp;</s>
  <s xml:id="echoid-s7786" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s7787" xml:space="preserve">CE) dico punctum Z ad oppoſitas hyperbolas, conſimili modo <lb/>determinabiles, exiſtere. </s>
  <s xml:id="echoid-s7788" xml:space="preserve">enimverò factis (ad utramque rectæ CA <lb/>partem) angulis ſemirectis ACP; </s>
  <s xml:id="echoid-s7789" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7790" xml:space="preserve">(ab ipſarum CP cum AE <lb/>occurſibus) ductis rectis RT, SV ad CE parallelis, punctum X <lb/>extra limites TV neceſſariò conſiſtet (etenim ubivis intra TV ductâ <lb/>LIP ad CE parallelâ, erit LI &amp;</s>
  <s xml:id="echoid-s7791" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s7792" xml:space="preserve">LP, ideóque nulla par ipſi LI <lb/>angulo AL I ſubtendi poteſt; </s>
  <s xml:id="echoid-s7793" xml:space="preserve">id quod extra terminos hoſce nil pro-<lb/>hibet fieri) erit jam TV axis, &amp; </s>
  <s xml:id="echoid-s7794" xml:space="preserve">C focus hyperbolarum. </s>
  <s xml:id="echoid-s7795" xml:space="preserve">Fiant <lb/>enim omnia, quæ in caſu præcedente; </s>
  <s xml:id="echoid-s7796" xml:space="preserve">erítque rurſus hîc KH = <lb/>KV. </s>
  <s xml:id="echoid-s7797" xml:space="preserve">item ob AV. </s>
  <s xml:id="echoid-s7798" xml:space="preserve">AT :</s>
  <s xml:id="echoid-s7799" xml:space="preserve">: CV. </s>
  <s xml:id="echoid-s7800" xml:space="preserve">DV; </s>
  <s xml:id="echoid-s7801" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7802" xml:space="preserve">(inversè componendo)</s>
</p>
<div xml:id="echoid-div200" type="float" level="2" n="2">
<note position="left" xlink:label="note-0136-01" xlink:href="note-0136-01a" xml:space="preserve">Fig. 191.</note>
</div>
<pb o="119" file="0137" n="137" rhead=""/>
<p>
  <s xml:id="echoid-s7803" xml:space="preserve">AV. </s>
  <s xml:id="echoid-s7804" xml:space="preserve">TV :</s>
  <s xml:id="echoid-s7805" xml:space="preserve">: CV. </s>
  <s xml:id="echoid-s7806" xml:space="preserve">CD; </s>
  <s xml:id="echoid-s7807" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7808" xml:space="preserve">conſequentes ſubduplandò, dividendó-<lb/>que AK. </s>
  <s xml:id="echoid-s7809" xml:space="preserve">KV :</s>
  <s xml:id="echoid-s7810" xml:space="preserve">: KV. </s>
  <s xml:id="echoid-s7811" xml:space="preserve">KD :</s>
  <s xml:id="echoid-s7812" xml:space="preserve">: KV. </s>
  <s xml:id="echoid-s7813" xml:space="preserve">CK. </s>
  <s xml:id="echoid-s7814" xml:space="preserve">vel AK. </s>
  <s xml:id="echoid-s7815" xml:space="preserve">KH :</s>
  <s xml:id="echoid-s7816" xml:space="preserve">: KH. </s>
  <s xml:id="echoid-s7817" xml:space="preserve">CK; <lb/></s>
  <s xml:id="echoid-s7818" xml:space="preserve">hoc eſt HN (KX). </s>
  <s xml:id="echoid-s7819" xml:space="preserve">NG :</s>
  <s xml:id="echoid-s7820" xml:space="preserve">: KH. </s>
  <s xml:id="echoid-s7821" xml:space="preserve">CK; </s>
  <s xml:id="echoid-s7822" xml:space="preserve">quare CK x KX = KH <lb/>x NG. </s>
  <s xml:id="echoid-s7823" xml:space="preserve">eſt autem XZq = CZq - CXq = XGq - CXq = <lb/>NGq + KHq - 2 NG x KH: </s>
  <s xml:id="echoid-s7824" xml:space="preserve">- KXq - CKq + 2 CK <lb/>x KX = NGq + KHq - KXq - CKq. </s>
  <s xml:id="echoid-s7825" xml:space="preserve">Sumatur K ζ = KX, <lb/>diſcurſúmque ſimilem adhibendo liquebit fore ξζ = XZ; </s>
  <s xml:id="echoid-s7826" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7827" xml:space="preserve">ideo <lb/>Dζ = CZ. </s>
  <s xml:id="echoid-s7828" xml:space="preserve">unde Cζ - Dζ (DZ - CZ) = Cζ - CZ = <lb/>ξγ - XG = 2 KH = TV. </s>
  <s xml:id="echoid-s7829" xml:space="preserve">quare manifeſtum eſt _cnrvas_ TZ, <lb/>Vζ eſſe _Hyperbolas,_ quarum axis TV, foci C,D.</s>
  <s xml:id="echoid-s7830" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7831" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s7832" xml:space="preserve">Tertiò demùm, ſit angulus CAE ſemirectus (vel CA = <lb/>
<anchor type="note" xlink:label="note-0137-01a" xlink:href="note-0137-01"/>
CE) erit tum punctum Z ad parabolam; </s>
  <s xml:id="echoid-s7833" xml:space="preserve">quæ itidem ita determina-<lb/>tur. </s>
  <s xml:id="echoid-s7834" xml:space="preserve">Fiat angulus ACP ſemirectus, &amp; </s>
  <s xml:id="echoid-s7835" xml:space="preserve">ab ipſarum AE, CP in-<lb/>terſectione R ducatur RT ad CE parallela; </s>
  <s xml:id="echoid-s7836" xml:space="preserve">erit T_Vertex_, atque C <lb/>_Focus Parabolæ._ </s>
  <s xml:id="echoid-s7837" xml:space="preserve">id quod ex bene nota ſectionis hujus proprietate con-<lb/>ſtat; </s>
  <s xml:id="echoid-s7838" xml:space="preserve">qua ſcilicet eſt TA = TR = TC (ob angulos TAR, <lb/>TCR ſemirectos) &amp; </s>
  <s xml:id="echoid-s7839" xml:space="preserve">AX = XG = CZ.</s>
  <s xml:id="echoid-s7840" xml:space="preserve"/>
</p>
<div xml:id="echoid-div201" type="float" level="2" n="3">
<note position="right" xlink:label="note-0137-01" xlink:href="note-0137-01a" xml:space="preserve">Fig. 192.</note>
</div>
<p>
  <s xml:id="echoid-s7841" xml:space="preserve">V. </s>
  <s xml:id="echoid-s7842" xml:space="preserve">Manifeſtum eſt verò rectam AE ſectiones has ad E contingere. <lb/></s>
  <s xml:id="echoid-s7843" xml:space="preserve">quia nempe perpetuò major eſt CZ (vel XG) ordinatâ XZ; </s>
  <s xml:id="echoid-s7844" xml:space="preserve">adeó-<lb/>que puncta G extra cuŕvas unaquæque jacent hoc eſt tota AG extra <lb/>illas cadit.</s>
  <s xml:id="echoid-s7845" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7846" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s7847" xml:space="preserve">Hiſce præſtratis: </s>
  <s xml:id="echoid-s7848" xml:space="preserve">_Eſto Circulare ſpeculum_ MBND, cen-<lb/>
<anchor type="note" xlink:label="note-0137-02a" xlink:href="note-0137-02"/>
trum habens C; </s>
  <s xml:id="echoid-s7849" xml:space="preserve">cui exponatur recta quæpiam F α G; </s>
  <s xml:id="echoid-s7850" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7851" xml:space="preserve">huic per-<lb/>pendicularis ſit recta C α; </s>
  <s xml:id="echoid-s7852" xml:space="preserve">quam ad parte@ averſas ſumpta CA, ad-<lb/>æquet. </s>
  <s xml:id="echoid-s7853" xml:space="preserve">Sit etiam CE ad CA perpendicularis, ac æqualis qua-<lb/>dranti diametri BD; </s>
  <s xml:id="echoid-s7854" xml:space="preserve">connexáque recta AE producatur utcunque. <lb/></s>
  <s xml:id="echoid-s7855" xml:space="preserve">ſumpto jam in recta F α G puncto quolibet F, connectatur FC, &amp; </s>
  <s xml:id="echoid-s7856" xml:space="preserve"><lb/>radiationis ab F in ipſa FC limes, ſeu _focus_, ſit Z; </s>
  <s xml:id="echoid-s7857" xml:space="preserve">ac per Z du-<lb/>catur ZX ad AC perpendicularis, ipſi AE occurrens in H; </s>
  <s xml:id="echoid-s7858" xml:space="preserve">dico <lb/>fore XH parem ipſi CZ.</s>
  <s xml:id="echoid-s7859" xml:space="preserve"/>
</p>
<div xml:id="echoid-div202" type="float" level="2" n="4">
<note position="right" xlink:label="note-0137-02" xlink:href="note-0137-02a" xml:space="preserve">Fig. 193.</note>
</div>
<p>
  <s xml:id="echoid-s7860" xml:space="preserve">Nam (è jam antè monſtratis) eſt FC. </s>
  <s xml:id="echoid-s7861" xml:space="preserve">CZ :</s>
  <s xml:id="echoid-s7862" xml:space="preserve">: FM. </s>
  <s xml:id="echoid-s7863" xml:space="preserve">MZ (hoc eſt) <lb/>:</s>
  <s xml:id="echoid-s7864" xml:space="preserve">: FC - CB. </s>
  <s xml:id="echoid-s7865" xml:space="preserve">CB - CZ. </s>
  <s xml:id="echoid-s7866" xml:space="preserve">hinc erit α C. </s>
  <s xml:id="echoid-s7867" xml:space="preserve">CX (AC. </s>
  <s xml:id="echoid-s7868" xml:space="preserve">CX) :</s>
  <s xml:id="echoid-s7869" xml:space="preserve">: <lb/>FC - CB. </s>
  <s xml:id="echoid-s7870" xml:space="preserve">CB - CZ. </s>
  <s xml:id="echoid-s7871" xml:space="preserve">quare (ducendo in ſe extrema, ac media) <lb/>erit AC x CB - AC x CZ = CX x FC - CX x CB. </s>
  <s xml:id="echoid-s7872" xml:space="preserve">hoc <lb/>eſt (ipſi CX x FC ſubſtituendo AC x CZ, propter α C. </s>
  <s xml:id="echoid-s7873" xml:space="preserve">CX :</s>
  <s xml:id="echoid-s7874" xml:space="preserve">: <lb/>FC. </s>
  <s xml:id="echoid-s7875" xml:space="preserve">CZ) erit AC x CB - AC x CZ = AC x CZ - CX x <lb/>CB. </s>
  <s xml:id="echoid-s7876" xml:space="preserve">tranſponendóque AC x CB + CX x CB = 2 AC x CZ.</s>
  <s xml:id="echoid-s7877" xml:space="preserve">
<pb o="120" file="0138" n="138" rhead=""/>
hoc eſt AX x CB = 2 AC x CZ; </s>
  <s xml:id="echoid-s7878" xml:space="preserve">vel 2 AX x CE = 2 AC <lb/>x CZ; </s>
  <s xml:id="echoid-s7879" xml:space="preserve">unde AX. </s>
  <s xml:id="echoid-s7880" xml:space="preserve">AC :</s>
  <s xml:id="echoid-s7881" xml:space="preserve">: CZ. </s>
  <s xml:id="echoid-s7882" xml:space="preserve">CE; </s>
  <s xml:id="echoid-s7883" xml:space="preserve">hoc eſt XH. </s>
  <s xml:id="echoid-s7884" xml:space="preserve">CE :</s>
  <s xml:id="echoid-s7885" xml:space="preserve">: CZ. <lb/></s>
  <s xml:id="echoid-s7886" xml:space="preserve">CE. </s>
  <s xml:id="echoid-s7887" xml:space="preserve">quapropter eſt XH = CZ : </s>
  <s xml:id="echoid-s7888" xml:space="preserve">Quod E. </s>
  <s xml:id="echoid-s7889" xml:space="preserve">D.</s>
  <s xml:id="echoid-s7890" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7891" xml:space="preserve">Quoad radiationem ad partes concavas, planè ſimilis eſt diſcurſus. <lb/></s>
  <s xml:id="echoid-s7892" xml:space="preserve">examinetis ipſi, peto.</s>
  <s xml:id="echoid-s7893" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7894" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s7895" xml:space="preserve">Exhinc evidenter liquet, ſi fuerit CA &amp;</s>
  <s xml:id="echoid-s7896" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s7897" xml:space="preserve">CE; </s>
  <s xml:id="echoid-s7898" xml:space="preserve">quòd om-<lb/>
<anchor type="note" xlink:label="note-0138-01a" xlink:href="note-0138-01"/>
nes punctorum F limites, ſeu foci (quales Z) ad ellipſin exiſtunt; <lb/></s>
  <s xml:id="echoid-s7899" xml:space="preserve">cujus _focus_ C, &amp; </s>
  <s xml:id="echoid-s7900" xml:space="preserve">cujus _axis_ TV è præmiſſis, non uno modo, deter-<lb/>minatur. </s>
  <s xml:id="echoid-s7901" xml:space="preserve">item ſi CA = CE, limites Z ad parabolam conſiſtent <lb/>cujus _focus_ C, _axis_ CT = {1/2} CE, _vertex_ T. </s>
  <s xml:id="echoid-s7902" xml:space="preserve">denuò, ſi CA &amp;</s>
  <s xml:id="echoid-s7903" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s7904" xml:space="preserve">CB, <lb/>puncta Z ad _h@perbolas eſſe conſtat_, quarum itidem _focus_ C; </s>
  <s xml:id="echoid-s7905" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7906" xml:space="preserve">_axis_ <lb/>TV facilè de modò (vel alibi) dictis reperitur; </s>
  <s xml:id="echoid-s7907" xml:space="preserve">cunctarum verò ſecti-<lb/>onum _Parameter_ ipſi CB æquatur.</s>
  <s xml:id="echoid-s7908" xml:space="preserve"/>
</p>
<div xml:id="echoid-div203" type="float" level="2" n="5">
<note position="left" xlink:label="note-0138-01" xlink:href="note-0138-01a" xml:space="preserve">Fig. 193, <lb/>194, 195.</note>
</div>
<p>
  <s xml:id="echoid-s7909" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s7910" xml:space="preserve">Hinc in ſingulis reſpectivè caſibus, ejuſmodi _ſectiones co-_ <lb/>_nicæ_ ſunt rectarum F α G abſolutæ imagines; </s>
  <s xml:id="echoid-s7911" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s7912" xml:space="preserve">eædem veræ <lb/>ſunt imagines ad oculum relatæ in ſpeculi centro conſtitutum; </s>
  <s xml:id="echoid-s7913" xml:space="preserve">ex re-<lb/>flectione ſcilicet ad concavas ſpeculi partes effectæ quæ ſolæ oculo <lb/>ſic poſito conſpicuæ ſunt.</s>
  <s xml:id="echoid-s7914" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7915" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s7916" xml:space="preserve">Patet autem ſirecta F α G infinitè diſtet, quòd _ellipſis_ in _cir-_ <lb/>_culum_ abit. </s>
  <s xml:id="echoid-s7917" xml:space="preserve">utì quoque ſi F α G per centrum tranſeat, quòd _hyperbolæ_ <lb/>iſtæ in rectam lineam degenerant.</s>
  <s xml:id="echoid-s7918" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7919" xml:space="preserve">X. </s>
  <s xml:id="echoid-s7920" xml:space="preserve">Subnotetur etiam in caſu quum _imago fit hyperbolica_, quod <lb/>_hyperbolæ_ YTY pars YEEY, neque non tota ζ V ζ ad circuli partes <lb/>MBN pertinent; </s>
  <s xml:id="echoid-s7921" xml:space="preserve">(nempe ſi centro C per E deſcriptus circulus ipſam <lb/>FG interſecet punctis K, tota hyperbola ζ V ζ rectam interceptam KK <lb/>referet; </s>
  <s xml:id="echoid-s7922" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7923" xml:space="preserve">hyperbolicæ lineæ alterius pars ſuperior YEEY quod <lb/>reliquum eſt repræ ſentabit hinc indè protenſæ rectæ FG) pars autem <lb/>ETE ad partem concavam MDN ſpectat. </s>
  <s xml:id="echoid-s7924" xml:space="preserve">id quod ſuffecerit ad-<lb/>monitum.</s>
  <s xml:id="echoid-s7925" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7926" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s7927" xml:space="preserve">Et hæc quidem de rectæ FAG imaginibus abſolutis; </s>
  <s xml:id="echoid-s7928" xml:space="preserve">è qui-<lb/>
<anchor type="note" xlink:label="note-0138-02a" xlink:href="note-0138-02"/>
bus commodius de relatis judicum fiet. </s>
  <s xml:id="echoid-s7929" xml:space="preserve">ſit, inſtantiæ loco, oculus O, <lb/>ad quem (convexis è partibus) ab F, &amp; </s>
  <s xml:id="echoid-s7930" xml:space="preserve">G reflectantur OMK, <lb/>ON L; </s>
  <s xml:id="echoid-s7931" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7932" xml:space="preserve">ſit ellipſis ZVYT abſoluta (qualem modò definivimus) <lb/>rectæ FAG imago; </s>
  <s xml:id="echoid-s7933" xml:space="preserve">quam ductæ FC, GC punctis Z, Y ſecent. <lb/></s>
  <s xml:id="echoid-s7934" xml:space="preserve">itaque punctorum F, G imagines ad O relatæ (puta φ, &amp;</s>
  <s xml:id="echoid-s7935" xml:space="preserve">γ) extra
<pb o="121" file="0139" n="139" rhead=""/>
ellipſin jacent. </s>
  <s xml:id="echoid-s7936" xml:space="preserve">Nam punctum K inter F &amp; </s>
  <s xml:id="echoid-s7937" xml:space="preserve">Z; </s>
  <s xml:id="echoid-s7938" xml:space="preserve">ac punctum φ inter <lb/>O, &amp; </s>
  <s xml:id="echoid-s7939" xml:space="preserve">K; </s>
  <s xml:id="echoid-s7940" xml:space="preserve">nec non punctum L inter G, &amp; </s>
  <s xml:id="echoid-s7941" xml:space="preserve">Y; </s>
  <s xml:id="echoid-s7942" xml:space="preserve">atque punctum γ in-<lb/>ter O, &amp; </s>
  <s xml:id="echoid-s7943" xml:space="preserve">L cadunt. </s>
  <s xml:id="echoid-s7944" xml:space="preserve">imaginis itaque φαγ figura ad ellipticam accedit; <lb/></s>
  <s xml:id="echoid-s7945" xml:space="preserve">eâ tamen aliquanto planior &amp; </s>
  <s xml:id="echoid-s7946" xml:space="preserve">compreſſior. </s>
  <s xml:id="echoid-s7947" xml:space="preserve">non diſſimili ratione quo-<lb/>ad imagines ad concava factas, &amp; </s>
  <s xml:id="echoid-s7948" xml:space="preserve">quoad cæteros caſus inſtituetur <lb/>judicium. </s>
  <s xml:id="echoid-s7949" xml:space="preserve">tædii plenum eſſet omnia ſingillatim percenſere. </s>
  <s xml:id="echoid-s7950" xml:space="preserve">quinetiam <lb/>ê præmiſſis luculentè conſtat quo pacto linea φαγ præcisè deſcribatur, <lb/>punctatim utique. </s>
  <s xml:id="echoid-s7951" xml:space="preserve">circa refractiones paria veniunt præſtanda; </s>
  <s xml:id="echoid-s7952" xml:space="preserve">poſt-<lb/>quam tamen paullùm reſpiravero; </s>
  <s xml:id="echoid-s7953" xml:space="preserve">nunc enim verbo quidem pauca, <lb/>rei qualitatem, ſtudiúmque demonſtrandis iſtis impenſum reſpectan-<lb/>do, ſatìs fortaſſe multa videor tradidiſſe.</s>
  <s xml:id="echoid-s7954" xml:space="preserve">‖</s>
</p>
<div xml:id="echoid-div204" type="float" level="2" n="6">
<note position="left" xlink:label="note-0138-02" xlink:href="note-0138-02a" xml:space="preserve">Fig. 196.</note>
</div>
</div>
<div xml:id="echoid-div206" type="section" level="1" n="25">
<head xml:id="echoid-head28" xml:space="preserve"><emph style="sc">Lect.</emph> XVIII.</head>
<p>
  <s xml:id="echoid-s7955" xml:space="preserve">I.</s>
  <s xml:id="echoid-s7956" xml:space="preserve">P_Ropoſitum eſt jam nobis rectæ lineæ ex refractione prognatas. </s>
  <s xml:id="echoid-s7957" xml:space="preserve">ad_ <lb/>_circulum imagines aeſignare_; </s>
  <s xml:id="echoid-s7958" xml:space="preserve">nempe primùm abſolutas; <lb/></s>
  <s xml:id="echoid-s7959" xml:space="preserve">quorſum hoc ſpectat I<unsure/> heorema:</s>
  <s xml:id="echoid-s7960" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s7961" xml:space="preserve">In circulum (e. </s>
  <s xml:id="echoid-s7962" xml:space="preserve">g. </s>
  <s xml:id="echoid-s7963" xml:space="preserve">medii denſioris) refractivum MBND radiet <lb/>recta FAG; </s>
  <s xml:id="echoid-s7964" xml:space="preserve">huic verò perpendicularis ſit recta CA (circuli cen-<lb/>
<anchor type="note" xlink:label="note-0139-01a" xlink:href="note-0139-01"/>
trum C permeans) tum in recta FG ſumpto liberè puncto F ducatur <lb/>recta FC; </s>
  <s xml:id="echoid-s7965" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7966" xml:space="preserve">in hac ſit punctum Z limes (qualem anteà fiximus) <lb/>radiationis à puncto F; </s>
  <s xml:id="echoid-s7967" xml:space="preserve">ſit autem ZX ad AC normalis. </s>
  <s xml:id="echoid-s7968" xml:space="preserve">porrò fiat <lb/>CA. </s>
  <s xml:id="echoid-s7969" xml:space="preserve">CR :</s>
  <s xml:id="echoid-s7970" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s7971" xml:space="preserve">R; </s>
  <s xml:id="echoid-s7972" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s7973" xml:space="preserve">AR. </s>
  <s xml:id="echoid-s7974" xml:space="preserve">CB :</s>
  <s xml:id="echoid-s7975" xml:space="preserve">: CR. </s>
  <s xml:id="echoid-s7976" xml:space="preserve">CE (ponatur autem <lb/>CE ad XZ parallela) tum connexa RE cum ipſa XZ conveniat in <lb/>H. </s>
  <s xml:id="echoid-s7977" xml:space="preserve">dico fore XH = CZ.</s>
  <s xml:id="echoid-s7978" xml:space="preserve"/>
</p>
<div xml:id="echoid-div206" type="float" level="2" n="1">
<note position="right" xlink:label="note-0139-01" xlink:href="note-0139-01a" xml:space="preserve">Fig. 197.</note>
</div>
<p>
  <s xml:id="echoid-s7979" xml:space="preserve">Nam (è præmonſtratis) eſt FC x MZ. </s>
  <s xml:id="echoid-s7980" xml:space="preserve">FM x CZ :</s>
  <s xml:id="echoid-s7981" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s7982" xml:space="preserve">R:</s>
  <s xml:id="echoid-s7983" xml:space="preserve">: <lb/>CA. </s>
  <s xml:id="echoid-s7984" xml:space="preserve">CR. </s>
  <s xml:id="echoid-s7985" xml:space="preserve">hoceſt FC x CM + FC x CZ. </s>
  <s xml:id="echoid-s7986" xml:space="preserve">FC x CZ - CM <lb/>x CZ :</s>
  <s xml:id="echoid-s7987" xml:space="preserve">: CA. </s>
  <s xml:id="echoid-s7988" xml:space="preserve">CR. </s>
  <s xml:id="echoid-s7989" xml:space="preserve">quare (ducendo in ſe extrema, mediáque) <lb/>eſt FC x CM x CR + FC x CZ x CR = FC x CZ x CA -<lb/>CM x CZ x CA = FC x CZ x CA - CM x FC x CX (quoniam ſcilicet eſt
<pb o="122" file="0140" n="140" rhead=""/>
CZ. </s>
  <s xml:id="echoid-s7990" xml:space="preserve">FC :</s>
  <s xml:id="echoid-s7991" xml:space="preserve">: CX. </s>
  <s xml:id="echoid-s7992" xml:space="preserve">CA; </s>
  <s xml:id="echoid-s7993" xml:space="preserve">adeóque CZ x CA = FC x CX). </s>
  <s xml:id="echoid-s7994" xml:space="preserve">qua-<lb/>propter (elidendo FC) eſt CM x CR + CZ x CR = CZ x <lb/>CA - CM x CX; </s>
  <s xml:id="echoid-s7995" xml:space="preserve">tranſponendóque CM x CR + CM x <lb/>CX = CZ x CA - CZ x CR. </s>
  <s xml:id="echoid-s7996" xml:space="preserve">hoc eſt CM x RX = CZ x <lb/>AR, quare (ad analogiſmum redigendo) eſt AR. </s>
  <s xml:id="echoid-s7997" xml:space="preserve">CM :</s>
  <s xml:id="echoid-s7998" xml:space="preserve">: RX. <lb/></s>
  <s xml:id="echoid-s7999" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s8000" xml:space="preserve">hoc eſt CR. </s>
  <s xml:id="echoid-s8001" xml:space="preserve">CE :</s>
  <s xml:id="echoid-s8002" xml:space="preserve">: RX. </s>
  <s xml:id="echoid-s8003" xml:space="preserve">CZ. </s>
  <s xml:id="echoid-s8004" xml:space="preserve">hoc eſt RX. </s>
  <s xml:id="echoid-s8005" xml:space="preserve">XH :</s>
  <s xml:id="echoid-s8006" xml:space="preserve">: RX. </s>
  <s xml:id="echoid-s8007" xml:space="preserve"><lb/>CZ; </s>
  <s xml:id="echoid-s8008" xml:space="preserve">unde XH = CZ : </s>
  <s xml:id="echoid-s8009" xml:space="preserve">Quod E. </s>
  <s xml:id="echoid-s8010" xml:space="preserve">D.</s>
  <s xml:id="echoid-s8011" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s8012" xml:space="preserve">II. </s>
  <s xml:id="echoid-s8013" xml:space="preserve">Exhinc (&amp; </s>
  <s xml:id="echoid-s8014" xml:space="preserve">ex iis quæ circa _ſectiones conicas_ nuperrimè ſunt <lb/>oſtenſa) liquido conſectatur, ſi CR major fuerit quàm CE (vel <lb/>quod eódem recidit, AR major quam CB) quòd punctorum om-<lb/>nium F in recta FA G imagines abſolutæ (quales Z) ad _ellipſin_ con-<lb/>ſiftent, cujus _Focus_ C, cujúſque penitus determinandæ modum ſatìs <lb/>facilem tunc oſtendimus. </s>
  <s xml:id="echoid-s8015" xml:space="preserve">item ſi CR = CE, quòd imagines iſtæ ad <lb/>parabolam erunt; </s>
  <s xml:id="echoid-s8016" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8017" xml:space="preserve">denique, ſi CR &amp;</s>
  <s xml:id="echoid-s8018" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s8019" xml:space="preserve">CE, quòd eædem in hy-<lb/>perbolis oppoſitis reperientur; </s>
  <s xml:id="echoid-s8020" xml:space="preserve">quarum etiam ſectionum focus com-<lb/>munis eſt punctum C, &amp; </s>
  <s xml:id="echoid-s8021" xml:space="preserve">quarum axes deſignandi modum reliquáque <lb/>circa ipſas præſertim advertenda declaravimus. </s>
  <s xml:id="echoid-s8022" xml:space="preserve">(Nempe, ſi rectæ <lb/>CP cum ipſa CA ſemirectos conſtituant angulos; </s>
  <s xml:id="echoid-s8023" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8024" xml:space="preserve">hæ rectam RE <lb/>interſecent ad puncta S, indéque demittantur ad A@ perpendiculares <lb/>ST, SV, erunt T, V axis termini, rectáque CE ſemi-parameter erit) <lb/>unde patet totius rectæ FAG ad infinitum protenſæ abſolutam ima-<lb/>ginem (quin &amp; </s>
  <s xml:id="echoid-s8025" xml:space="preserve">illam, quæ ad oculum in centro O poſitum refertur) <lb/>aliquam eſſe dictarum conicarum, pro ſuo peculiari ſitu hanc vel illam <lb/>reſpectivè.</s>
  <s xml:id="echoid-s8026" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s8027" xml:space="preserve">III. </s>
  <s xml:id="echoid-s8028" xml:space="preserve">Adnotari porrò debet in iſto caſu, _ſectionis ellipticæ_ (quinetiam <lb/>
<anchor type="note" xlink:label="note-0140-01a" xlink:href="note-0140-01"/>
&amp; </s>
  <s xml:id="echoid-s8029" xml:space="preserve">_parabolicæ_) TE Z partem anticam TE ad concavas circuli partes <lb/>LDL ſpectare; </s>
  <s xml:id="echoid-s8030" xml:space="preserve">ſicuti poſtica EY ad convexas MB N pertinet. </s>
  <s xml:id="echoid-s8031" xml:space="preserve">in <lb/>hoc autem altero tota _byperbola_ ZVZ, nec non _hyperbotæ_ ETE pars <lb/>(infra ECE) YEEY ad partem circuli convexam referri debent <lb/>(nempe ſi centro C, intervallo CE deſcriptus circulus rectam FG <lb/>ſecet punctis K, K; </s>
  <s xml:id="echoid-s8032" xml:space="preserve">hyperbola ZVZ rectam interceptam KK repræ-<lb/>ſentabit, ipſiúſque FG quod reliquum eſt hinc indè protenſum pars <lb/>YEEY referet) pars autem ſuperior ETE ad cavam circuli partem <lb/>LDL ſpectat.</s>
  <s xml:id="echoid-s8033" xml:space="preserve">‖ Semper autem (cúm hîc, tum ubique) intelligatur <lb/>ad utraſque propoſiti circuli partes ejuſdem generis refractionem effici, <lb/>ſeu ejuſdem ſpeciei medio radios incidere.</s>
  <s xml:id="echoid-s8034" xml:space="preserve"/>
</p>
<div xml:id="echoid-div207" type="float" level="2" n="2">
<note position="left" xlink:label="note-0140-01" xlink:href="note-0140-01a" xml:space="preserve">Fig. 198.</note>
</div>
<p>
  <s xml:id="echoid-s8035" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s8036" xml:space="preserve">Ex his obiter naturæ, quam in oculi figura conſtruenda adhi-
<pb o="123" file="0141" n="141" rhead=""/>
buit, ſolertia quadantenus eluceſcere videatur, ſeu ratio quædam aſſig-<lb/>nari poſſit, cur oculi fundus _Sphær@idicam_ (aut ab hac non multum <lb/>abludentem) nacta ſit ſiguram. </s>
  <s xml:id="echoid-s8037" xml:space="preserve">quia nimirum illa planorum objecto-<lb/>rum modicè diſtantium (quibus in diſtinctiùs apprehendendis po-<lb/>tiſſimus verſatur uſus) excipiendis ſimulachris eſt accommodatiſſima. <lb/></s>
  <s xml:id="echoid-s8038" xml:space="preserve">Sed hoc παρεισθδκῶς.</s>
  <s xml:id="echoid-s8039" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s8040" xml:space="preserve">V. </s>
  <s xml:id="echoid-s8041" xml:space="preserve">In reliquis reſractionum caſibus paria fermè contingunt, quos <lb/>ideò tacitus præterlabi poſſem; </s>
  <s xml:id="echoid-s8042" xml:space="preserve">at minuendo veſtro labori, ſeu quò <lb/>clariùs &amp; </s>
  <s xml:id="echoid-s8043" xml:space="preserve">promptiùs de iis conſtet, non gravabor &amp; </s>
  <s xml:id="echoid-s8044" xml:space="preserve">illos vobis ob <lb/>oculos ponere: </s>
  <s xml:id="echoid-s8045" xml:space="preserve">nempe</s>
</p>
<p>
  <s xml:id="echoid-s8046" xml:space="preserve">Rarioris medii circulo MBN objiciatur recta FAG, cui normalis <lb/>
<anchor type="note" xlink:label="note-0141-01a" xlink:href="note-0141-01"/>
CA; </s>
  <s xml:id="echoid-s8047" xml:space="preserve">sitque punctum Z puncti cujuſvis F, in FG ſumpti, imago <lb/>abſoluta; </s>
  <s xml:id="echoid-s8048" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8049" xml:space="preserve">ZX ad CA perpendicularis; </s>
  <s xml:id="echoid-s8050" xml:space="preserve">ac CA. </s>
  <s xml:id="echoid-s8051" xml:space="preserve">CR:</s>
  <s xml:id="echoid-s8052" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s8053" xml:space="preserve">R; <lb/></s>
  <s xml:id="echoid-s8054" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8055" xml:space="preserve">RA. </s>
  <s xml:id="echoid-s8056" xml:space="preserve">CB:</s>
  <s xml:id="echoid-s8057" xml:space="preserve">: RC. </s>
  <s xml:id="echoid-s8058" xml:space="preserve">CE; </s>
  <s xml:id="echoid-s8059" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8060" xml:space="preserve">ipſi RE connexæ occurrat XZ pro-<lb/>tracta ad H; </s>
  <s xml:id="echoid-s8061" xml:space="preserve">eritque rurſus XH = CZ.</s>
  <s xml:id="echoid-s8062" xml:space="preserve"/>
</p>
<div xml:id="echoid-div208" type="float" level="2" n="3">
<note position="right" xlink:label="note-0141-01" xlink:href="note-0141-01a" xml:space="preserve">Fig. 199.</note>
</div>
<p>
  <s xml:id="echoid-s8063" xml:space="preserve">Nam eſt CA. </s>
  <s xml:id="echoid-s8064" xml:space="preserve">CR:</s>
  <s xml:id="echoid-s8065" xml:space="preserve">: (FC x MZ. </s>
  <s xml:id="echoid-s8066" xml:space="preserve">FM x CZ:</s>
  <s xml:id="echoid-s8067" xml:space="preserve">:) FC x CZ <lb/>- FC x CM. </s>
  <s xml:id="echoid-s8068" xml:space="preserve">FC x CZ- CM x CZ. </s>
  <s xml:id="echoid-s8069" xml:space="preserve">quare CR x FC x CZ <lb/>- CR x FC x CM = CA x FC x CZ - CA x CM x CZ = CA x FC x CZ - FC x CM x CX. </s>
  <s xml:id="echoid-s8070" xml:space="preserve">ac indè CR x CZ-<lb/>CR x CM = CA x CZ - CM x CX. </s>
  <s xml:id="echoid-s8071" xml:space="preserve">tranſponendóque CR x <lb/>CZ - CA x CZ = CR x CM - CX x CM; </s>
  <s xml:id="echoid-s8072" xml:space="preserve">hoc eſt RX. <lb/></s>
  <s xml:id="echoid-s8073" xml:space="preserve">CZ:</s>
  <s xml:id="echoid-s8074" xml:space="preserve">: AR. </s>
  <s xml:id="echoid-s8075" xml:space="preserve">CM:</s>
  <s xml:id="echoid-s8076" xml:space="preserve">: RC. </s>
  <s xml:id="echoid-s8077" xml:space="preserve">CE:</s>
  <s xml:id="echoid-s8078" xml:space="preserve">: RX. </s>
  <s xml:id="echoid-s8079" xml:space="preserve">XH. </s>
  <s xml:id="echoid-s8080" xml:space="preserve">quapropter eſt CZ = XH.</s>
  <s xml:id="echoid-s8081" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s8082" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s8083" xml:space="preserve">Hinc dilucidè rurſus apparet rectæ FA Gimaginem abſolu-<lb/>tam (vel ad oculum in centro C ſitum relatam) ſi RC &amp;</s>
  <s xml:id="echoid-s8084" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s8085" xml:space="preserve">CE, _el-_ <lb/>_lipticam_ fore; </s>
  <s xml:id="echoid-s8086" xml:space="preserve">ſin RC = CE, fore _parabolicam_ (quarum ſectio-<lb/>num pars anterior ETE ad convexam circuli refringentis partem <lb/>
<anchor type="note" xlink:label="note-0141-02a" xlink:href="note-0141-02"/>
MBN pertinet, poſterior YEEY ad cavam LDL). </s>
  <s xml:id="echoid-s8087" xml:space="preserve">Quòd ſi <lb/>fuerit RC &amp;</s>
  <s xml:id="echoid-s8088" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s8089" xml:space="preserve">RE, _ejus image hyperbolica erit_; </s>
  <s xml:id="echoid-s8090" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8091" xml:space="preserve">quidem _hyperbolæ_ <lb/>YTY pars ſuperior ETE ad circuli partem NBN referenda eſt; <lb/></s>
  <s xml:id="echoid-s8092" xml:space="preserve">pars autem inferior YEEY unà cum tota hyperbola ζ V ζ ad partes <lb/>concavas LDL pertinebit. </s>
  <s xml:id="echoid-s8093" xml:space="preserve">nempe ſi fuerint rectæ CK æquales ipſi <lb/>CE, tota hyperbola ζ V ζ interceptam punctis K rectæ FG portio-<lb/>nem referet, ejúſque quod hinc indè protenſum ſupereſt ab ipſa YEEY <lb/>repræſentabitur.</s>
  <s xml:id="echoid-s8094" xml:space="preserve"/>
</p>
<div xml:id="echoid-div209" type="float" level="2" n="4">
<note position="right" xlink:label="note-0141-02" xlink:href="note-0141-02a" xml:space="preserve">Fig. 200.</note>
</div>
<p>
  <s xml:id="echoid-s8095" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s8096" xml:space="preserve">Porrò, quoad omnes hoſce caſus animadvertere licet poſſe <lb/>ſectionem eandem conicam innumeris rectis lineis ad diverſos circulos
<pb o="124" file="0142" n="142" rhead=""/>
concentricos expoſitis repræſentandis inſervire. </s>
  <s xml:id="echoid-s8097" xml:space="preserve">nimirum in caſu po-<lb/>ſtremo, ſi reliquis ſtantibus punctum A indeterminatum ponatur, ni-<lb/>hilominus hyperbolæ ζυζ, YTY rectas FAG repræſentabunt ad <lb/>circulos, quorum ſemidiametri CB ipſis AI ſingulæ reſpectivæ <lb/>ſingulis æ quantur, modò ſemper intelligatur eſſe CA. </s>
  <s xml:id="echoid-s8098" xml:space="preserve">CR:</s>
  <s xml:id="echoid-s8099" xml:space="preserve">: I. </s>
  <s xml:id="echoid-s8100" xml:space="preserve">R. <lb/></s>
  <s xml:id="echoid-s8101" xml:space="preserve">id quod ſatìs fuerit obiter admonuiſſe.</s>
  <s xml:id="echoid-s8102" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s8103" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s8104" xml:space="preserve">Ut &amp; </s>
  <s xml:id="echoid-s8105" xml:space="preserve">illud curſim innuiſſe ſuſſecerit, quòd ſicut à conicis <lb/>ſectionibus rectæ lineæ, ita viciſſim _conicæ ſectiones_ à rectis lineis ex <lb/>juſta congruos ad circulos inſlectione repræſentantur; </s>
  <s xml:id="echoid-s8106" xml:space="preserve">quos utique <lb/>non arduum videtur è præmiſſis deducere.</s>
  <s xml:id="echoid-s8107" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s8108" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s8109" xml:space="preserve">Ut &amp; </s>
  <s xml:id="echoid-s8110" xml:space="preserve">exindè _datâ conicâ ſe<unsure/>ctione_ circulus &amp; </s>
  <s xml:id="echoid-s8111" xml:space="preserve">recta facilè deſignan-<lb/>tur, ità ut conica rectam illam repræſentet ex inſlectione ad iſtum <lb/>circulum. </s>
  <s xml:id="echoid-s8112" xml:space="preserve">Nempe ſi à foco C ad axem CV applicetur normalis <lb/>CE; </s>
  <s xml:id="echoid-s8113" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8114" xml:space="preserve">recta ER ſectionem tangat ad E; </s>
  <s xml:id="echoid-s8115" xml:space="preserve">factoque CR. </s>
  <s xml:id="echoid-s8116" xml:space="preserve">CA:</s>
  <s xml:id="echoid-s8117" xml:space="preserve">: <lb/>R. </s>
  <s xml:id="echoid-s8118" xml:space="preserve">I; </s>
  <s xml:id="echoid-s8119" xml:space="preserve">ducatur per A recta AI ad CE parallela; </s>
  <s xml:id="echoid-s8120" xml:space="preserve">sitque CB = AI; <lb/></s>
  <s xml:id="echoid-s8121" xml:space="preserve">tum centro C per B ducatur circulus MBN, peractum erit nego-<lb/>tium.</s>
  <s xml:id="echoid-s8122" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s8123" xml:space="preserve">X. </s>
  <s xml:id="echoid-s8124" xml:space="preserve">Ex his tandem de imaginibus ad oculum ubicunque collocatum <lb/>relatis, quales illæ figuras ac ſitus obtinent, proclivius erit judicare. <lb/></s>
  <s xml:id="echoid-s8125" xml:space="preserve">ſcilicet eæ ſaltem unum (in recta per oculi, circulique reſringentis <lb/>centrum trajecta poſitum) commune cum abſolutis punctum habent; </s>
  <s xml:id="echoid-s8126" xml:space="preserve"><lb/>quoad reliqua vero reſpectiva puncta nonnihil ab his deſlectunt ad eas <lb/>partes, quas oculi ſitus peculiaris, &amp; </s>
  <s xml:id="echoid-s8127" xml:space="preserve">radiorum curſus exigunt; </s>
  <s xml:id="echoid-s8128" xml:space="preserve">id <lb/>quod facilius ſit in ſingulis caſibus qualiter eveniat perſpicere, quam <lb/>verbis univerſim explicare. </s>
  <s xml:id="echoid-s8129" xml:space="preserve">ſed enim unam rei declarandæ ſubjicie-<lb/>mus inſtantiam. </s>
  <s xml:id="echoid-s8130" xml:space="preserve">Ad oculum O refringantur ab F, &amp; </s>
  <s xml:id="echoid-s8131" xml:space="preserve">G radii FMO, <lb/>GNO; </s>
  <s xml:id="echoid-s8132" xml:space="preserve">ſit autem _ellipſis_ TZVY rectæ FG abſoluta imago, quam <lb/>connexæ FC, GC punctis Z, Y ſecent (ità quidem ut Z ſit puncti F, <lb/>
<anchor type="note" xlink:label="note-0142-01a" xlink:href="note-0142-01"/>
&amp; </s>
  <s xml:id="echoid-s8133" xml:space="preserve">Y puncti G imago abſoluta) enimverò, de ſupradictis colligitur <lb/>punctum K ſupra Z verſus C exiſtere; </s>
  <s xml:id="echoid-s8134" xml:space="preserve">quinetiam puncti F in recta <lb/>MO imaginem (puta φ) ultra FZ jacere. </s>
  <s xml:id="echoid-s8135" xml:space="preserve">Similiter puncti G ima-<lb/>go (γ) ſupra Y, ultráque GY ſita eſt. </s>
  <s xml:id="echoid-s8136" xml:space="preserve">unde conjectura ſiet de totius <lb/>imaginis φ α γ poſitione, ſeu figura ad _ellipticam_ accedente, qualis <lb/>in appoſita exhibetur figura; </s>
  <s xml:id="echoid-s8137" xml:space="preserve">quæ certè (quanquam haud abſque <lb/>nimiâ moleſtiâ) juxta theoriam ſuprà conſtabilitam accuratè poterit <lb/>delineari.</s>
  <s xml:id="echoid-s8138" xml:space="preserve"/>
</p>
<div xml:id="echoid-div210" type="float" level="2" n="5">
<note position="left" xlink:label="note-0142-01" xlink:href="note-0142-01a" xml:space="preserve">Fig. 201.</note>
</div>
<pb o="125" file="0143" n="143" rhead=""/>
<p>
  <s xml:id="echoid-s8139" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s8140" xml:space="preserve">Ità rectarum linearum ad ſphæricam ſuperſiciem ex inſlectione <lb/>quavis procreatas imagines qualitercunque liceat definire. </s>
  <s xml:id="echoid-s8141" xml:space="preserve">unde de <lb/>planarum quoque ſuperſicierum ad eandem repræſentationibus haud <lb/>difficilè ſtatuetur; </s>
  <s xml:id="echoid-s8142" xml:space="preserve">harum ſcilicet imagines abſolutæ _conoidum aut_ <lb/>_Sphæroidum Superſicies erunt_ è rectarum imaginibus reſpectivis circa <lb/>radiationum axes converſis progenitæ; </s>
  <s xml:id="echoid-s8143" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s8144" xml:space="preserve">relatæ quoque pla-<lb/>narum ſuperficierum imagines è rectarum imaginibus relatis ſimili <lb/>pacto progenerantur. </s>
  <s xml:id="echoid-s8145" xml:space="preserve">rem totam ipſi mentem aliquantillùm adver-<lb/>tentes perſpicietis; </s>
  <s xml:id="echoid-s8146" xml:space="preserve">me λεπτολογιας<unsure/> extremæ faſtidium capit.</s>
  <s xml:id="echoid-s8147" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s8148" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s8149" xml:space="preserve">Reſtare videtur, ut quomodò compoſitæ ſuperficies ſphæricæ <lb/>objectas repræſentant lineas diſpiciamus. </s>
  <s xml:id="echoid-s8150" xml:space="preserve">verum cum imagines indè <lb/>prognatæ ſint altioris gradûs lineæ, ab uſu notitiàque communi ſegre-<lb/>gatæ, atque proprietatibus intricatis præditæ; </s>
  <s xml:id="echoid-s8151" xml:space="preserve">nil aliud quàm operam <lb/>luderem iis deſudans extricandis. </s>
  <s xml:id="echoid-s8152" xml:space="preserve">illas itaque tranſiliam; </s>
  <s xml:id="echoid-s8153" xml:space="preserve">hoc com-<lb/>monens unicum, punctorum in illis aliquot principalium poſitiones è <lb/>præmonſtratis dignoſci, de cæteris commodiùs ex conjectura diju-<lb/>dicari.</s>
  <s xml:id="echoid-s8154" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s8155" xml:space="preserve">XIII Hæc ſunt, quæ circa partem _Opticæ_ præcipuè _Mathemati-_ <lb/>_cam_ dicenda mihi ſuggeſſit meditatio. </s>
  <s xml:id="echoid-s8156" xml:space="preserve">circa reliquas (quæ φυσικώτε{ρο}ι <lb/>ſunt, adeóqve ſæpiuſculè pro certis principiis plauſibiles conjecturas <lb/>venditare neceſſum habent) nihil ferè quicquam admodum veriſimile <lb/>ſuccurrit, à pervulgatis (ab iis, inquam, quæ _Keplerus, Sche@ner@s,_ <lb/>_Carteſius_, &amp; </s>
  <s xml:id="echoid-s8157" xml:space="preserve">poſt illos alii tradiderunt) alienum aut diverſum: </s>
  <s xml:id="echoid-s8158" xml:space="preserve">atqui <lb/>tacere malo, quàm toties oblatam cramben reponere. </s>
  <s xml:id="echoid-s8159" xml:space="preserve">proinde re-<lb/>
<anchor type="handwritten" xlink:label="hd-0143-01a" xlink:href="hd-0143-01"/>
ceptui cano; </s>
  <s xml:id="echoid-s8160" xml:space="preserve">nec ità tamen ut prorſus diſcedam, anteaquàm impro-<lb/>bam quandam diſſicultatem (pro ſinceritate quam &amp; </s>
  <s xml:id="echoid-s8161" xml:space="preserve">vobis &amp; </s>
  <s xml:id="echoid-s8162" xml:space="preserve">veritati <lb/>debeo minimè diſſimulandam) in medium protulero, quæ doctrinæ <lb/>noſtræ, hactenus inculcatæ, ſe objicit adverſam, ab ea ſaltem nul-<lb/>lam admittit ſolutionem. </s>
  <s xml:id="echoid-s8163" xml:space="preserve">illa, breviter, talis eſt: </s>
  <s xml:id="echoid-s8164" xml:space="preserve">_Lenti vel ſpeculo_ <lb/>
<anchor type="note" xlink:label="note-0143-01a" xlink:href="note-0143-01"/>
_cavo_ EBF exponatur viſibile punctum A, ità diſtans, ut radii ab A <lb/>manantes ex inflectione verſus axem AB cogantur; </s>
  <s xml:id="echoid-s8165" xml:space="preserve">si<unsure/>tque radiationis <lb/>limes (ſeu puncti A imago, qualem ſupra paſſim ſtatuimus) pun-<lb/>ctum Z; </s>
  <s xml:id="echoid-s8166" xml:space="preserve">inter hoc autem &amp; </s>
  <s xml:id="echoid-s8167" xml:space="preserve">inflectentis verticem B uſpiam poſitus <lb/>concipiatur oculus. </s>
  <s xml:id="echoid-s8168" xml:space="preserve">quæri jam poteſt, ubi loci debeat punctum A ap-<lb/>parere. </s>
  <s xml:id="echoid-s8169" xml:space="preserve">retrorſum ad punctum Z videri natura non fert (cùm omnis <lb/>impreſſio ſenſum aſſiciens proveniat a partibus A) ac experientia re-<lb/>clamat. </s>
  <s xml:id="echoid-s8170" xml:space="preserve">noſtris autem è placitis conſequi videtur ipſum, ad partes an-
<pb o="126" file="0144" n="144" rhead=""/>
ticas apparens, ab intervallo longiſſimè diſſito, (quod &amp; </s>
  <s xml:id="echoid-s8171" xml:space="preserve">maximum <lb/>ſenſibile quodvis intervallum quodammodò exſuperet) apparere. <lb/></s>
  <s xml:id="echoid-s8172" xml:space="preserve">cùm enim quò radiis minùs divergentibus attingitur objectum, eò <lb/>(ſecluſis utique prænotionibus, &amp; </s>
  <s xml:id="echoid-s8173" xml:space="preserve">præjudiciis) longiùs abeſſe ſentia-<lb/>tur; </s>
  <s xml:id="echoid-s8174" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8175" xml:space="preserve">quod parallelos ad oculum radios projicit, remotiſſimè poſi-<lb/>tum æſtimetur; </s>
  <s xml:id="echoid-s8176" xml:space="preserve">exigere ratio videtur, ut quod convergentibus radiis <lb/>apprehenditur, adhuc magìs, ſi ſieri poſſet, quoad apparentiam <lb/>elongetur. </s>
  <s xml:id="echoid-s8177" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s8178" xml:space="preserve">circa caſum hunc generatim inquiri poſſit, quidnam <lb/>omninò ſit, quod apparentem puncti A locum determinet, faciátque <lb/>quòd conſtanti ratione nunc propius, nunc remotius appareat; </s>
  <s xml:id="echoid-s8179" xml:space="preserve">cui <lb/>itidem dubio nihil quicquam ex hactenus dictorum _Analogia_ reſpon-<lb/>deri poſſe videtur, niſi debere punctum A perpetuò longiſſimè ſemo-<lb/>tum videri. </s>
  <s xml:id="echoid-s8180" xml:space="preserve">Verùm experientia ſecùs atteſtatur, illud pro diversâ oculi <lb/>inter puncta B, Z poſitione variè diſtans; </s>
  <s xml:id="echoid-s8181" xml:space="preserve">nunquam ſerè (ſiunquam) <lb/>longinquius ipſo A liberè ſpectato, ſubindè vero multo propinquius <lb/>adparere; </s>
  <s xml:id="echoid-s8182" xml:space="preserve">quinimò, quo oculum appellentes radii magìs convergunt <lb/>eò ſpeciem objecti propiùs accedere. </s>
  <s xml:id="echoid-s8183" xml:space="preserve">nempe, ſi puncto B admoveatur <lb/>_ocmlus,_ ſuo (ad lentem) ferè nativo in loco conſpicitur punctum A <lb/>(vel æquè diſtans, ad _ſpeculum_;) </s>
  <s xml:id="echoid-s8184" xml:space="preserve">ad O reductus oculus ejuſce ſpeciem <lb/>appropinquantem cernit; </s>
  <s xml:id="echoid-s8185" xml:space="preserve">ad P adhuc vicinius ipſum exiſtimat; </s>
  <s xml:id="echoid-s8186" xml:space="preserve">ac <lb/>ità ſenſim, donec alicubi tandem, velut ad Q, conſtituto oculo ob-<lb/>jectum ſummè propinquum apparens in meram conſuſionem incipiat <lb/>evaneſcere. </s>
  <s xml:id="echoid-s8187" xml:space="preserve">quæ ſanè cuncta rationibus atque decretis noſtris repug-<lb/>nare videntur, aut cum iis ſaltem parùm amicè conſpirant. </s>
  <s xml:id="echoid-s8188" xml:space="preserve">Neque <lb/>noſtram tantùm ſententiam pulſat hoc experimentum; </s>
  <s xml:id="echoid-s8189" xml:space="preserve">at ex æquo <lb/>cæteras quas nôrim omnes; </s>
  <s xml:id="echoid-s8190" xml:space="preserve">veterem imprimìs ac vulgatam; </s>
  <s xml:id="echoid-s8191" xml:space="preserve">noſtræ <lb/>præ reliquis aſſinem ità convellere videtur, ut ejus vi coactus doctiſſi-<lb/>mus _A. </s>
  <s xml:id="echoid-s8192" xml:space="preserve">Tacquetus_ iſti principio (cui penè ſoli totam inædificaverat <lb/>_Catoptricam_ ſuam) ceu inſido ac inconſtanti renunciarit, adeoque <lb/>ſuam ipſe doctrinam labefactârit; </s>
  <s xml:id="echoid-s8193" xml:space="preserve">id tamen, opinor, minimè facturus, <lb/>ſi rem totam inſpexiſſet penitiùs, atque difficultatis fundum attigiſſet. </s>
  <s xml:id="echoid-s8194" xml:space="preserve"><lb/>Apud me verò non ità pollet hæc, nec eoúſque præpollebit ulla diffi-<lb/>cultas, ut ab iis quæ manifeſtè rationi conſentanea video, diſcedam; </s>
  <s xml:id="echoid-s8195" xml:space="preserve"><lb/>præſertim quum ut hîc accidit, ejuſmodi difficultas in ſingularis cujus <lb/>piam casûs diſparitate fundetur. </s>
  <s xml:id="echoid-s8196" xml:space="preserve">nimirum in præſente caſu peculiare <lb/>quiddam, naturæ ſubtilitati involutum, deliteſcit, ægrè fortaſsìs, niſi <lb/>perfectius explorato videndi modo, detegendum. </s>
  <s xml:id="echoid-s8197" xml:space="preserve">circa quod nil, fateor, <lb/>hactenus excogitare potui, quod adblandiretur animo meo, nedum planè <lb/>ſatisfaceret. </s>
  <s xml:id="echoid-s8198" xml:space="preserve">Vobis itaque nodum hunc, utinam feliciore conatu, re-<lb/>ſolvendum committo. </s>
  <s xml:id="echoid-s8199" xml:space="preserve">Ità demum, _Auditores Optimi, Valeatis._</s>
  <s xml:id="echoid-s8200" xml:space="preserve"/>
</p>
<div xml:id="echoid-div211" type="float" level="2" n="6">
  <handwritten xlink:label="hd-0143-01" xlink:href="hd-0143-01a"/>
<note position="right" xlink:label="note-0143-01" xlink:href="note-0143-01a" xml:space="preserve">Fig. 202, <lb/>203,</note>
</div>
<pb o="127" file="0145" n="145"/>
</div>
<div xml:id="echoid-div213" type="section" level="1" n="26">
<head xml:id="echoid-head29" xml:space="preserve">ERRATA.</head>
<p>
  <s xml:id="echoid-s8201" xml:space="preserve">PAg. </s>
  <s xml:id="echoid-s8202" xml:space="preserve">3. </s>
  <s xml:id="echoid-s8203" xml:space="preserve">lin. </s>
  <s xml:id="echoid-s8204" xml:space="preserve">25 luce. </s>
  <s xml:id="echoid-s8205" xml:space="preserve">(præſente, leg. </s>
  <s xml:id="echoid-s8206" xml:space="preserve">luce (præſente p4, l 21, deſceptatur l diſceptatur <lb/>ib. </s>
  <s xml:id="echoid-s8207" xml:space="preserve">l 31, valeat, id l valeat id p5, l 31, toto l tota, p 6, l 23, dici l dicitur, p 11, <lb/>l 10, allas l alias, p 13, l 27, proximo l proximos, p 14, l 12, contraniſmum <lb/>l contraniſum, p 14, l 16, effectant l affectant, p 15, l 14, nobile l mobile, p 16, <lb/>l 20, in quæ l in iis quæ, p 17, l 3, ſubſtracto l ſubſtrato, p 17, l 5, citentur fig 10, <lb/>p 17, l 20, fig 12 l fig 11, p 18, l 11, fig 13 l fig 12, p 19, l 17, tranſmitti l tranſ-<lb/>mittit, p 22, l 21 πρθλεπτικῶς lπροληπτικῶς, p 23, l 6, incidentes (radios l inci-<lb/>dentes @adios, p 23, l 19, SB protracta lSB (protracta, p 28, ambages. </s>
  <s xml:id="echoid-s8208" xml:space="preserve">I.</s>
  <s xml:id="echoid-s8209" xml:space="preserve">De-<lb/>monſtratæ proſtant l ambages demonſtratæ proſtant. </s>
  <s xml:id="echoid-s8210" xml:space="preserve">I. </s>
  <s xml:id="echoid-s8211" xml:space="preserve">Ut in Parabola, p 29, <lb/>l 25, citentur fig 29, p 29 l 20, puncto divergentium tanqu@m l puncto divergen-<lb/>tium radiorum reflexi rurſus divergunt tanquam, p 29, l 32, ſuo l ſeu, p 32, l 10, <lb/>fig 34, 35 deleatur, &amp; </s>
  <s xml:id="echoid-s8212" xml:space="preserve">citentur ad lineam 31, p 33, l4, citentur fig 36, ibid. </s>
  <s xml:id="echoid-s8213" xml:space="preserve">l 6, <lb/>citentur fig 37 &amp; </s>
  <s xml:id="echoid-s8214" xml:space="preserve">38, p 38, l 28, I q. </s>
  <s xml:id="echoid-s8215" xml:space="preserve">R. </s>
  <s xml:id="echoid-s8216" xml:space="preserve">AB. </s>
  <s xml:id="echoid-s8217" xml:space="preserve">@ lIq. </s>
  <s xml:id="echoid-s8218" xml:space="preserve">R. </s>
  <s xml:id="echoid-s8219" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8220" xml:space="preserve">AB. </s>
  <s xml:id="echoid-s8221" xml:space="preserve">T p 40, l17, <lb/>Sc l Si, ib. </s>
  <s xml:id="echoid-s8222" xml:space="preserve">l33, quoquam l quaquam, p 41, l 24, abeoque l adeoque, p 45, l7, <lb/>reciſſimi l rectiſſimi, ib l8, propriores l propiores, p 47, l 24, reſignare l deſignare, <lb/>p 48, l1, Z 2, l2. </s>
  <s xml:id="echoid-s8223" xml:space="preserve">ib. </s>
  <s xml:id="echoid-s8224" xml:space="preserve">l 12, Nocetur ſi ſuerit HNP l Notetu ſi fuerit HN @, <lb/>p 57, l4, ejuſce l cn uſce, p 65, l4, exiſtimari l exiſtimare, ib. </s>
  <s xml:id="echoid-s8225" xml:space="preserve">l22, ipſe l eſſe, p 67, <lb/>l 13, interjaceret l interjacet, p 70, l 3, poſteribus l poſterioribus p 71, l 32, Ad-<lb/>verſatur l Advertatur, p 71, l 15, γ S γ v l γ S. </s>
  <s xml:id="echoid-s8226" xml:space="preserve">γ v. </s>
  <s xml:id="echoid-s8227" xml:space="preserve">p 78, l1, in eodem le<unsure/>o-<lb/>dem, p 89, l 25. </s>
  <s xml:id="echoid-s8228" xml:space="preserve">ratiori l rariori, p 90, l 17, expanſem l expenſam, p 95, l 14, <lb/>relect onibus l reflectioni@us, ib. </s>
  <s xml:id="echoid-s8229" xml:space="preserve">l 40, Sinus l Simus, p 96, l 3, ſim<unsure/>plicimè l ſimpli-<lb/>ciſſime, p 97, l8, quæſitam l quæſitum, p 105, l 28, Poſito l Poſitio, p 112, l 26, <lb/>admodum l ad modum.</s>
  <s xml:id="echoid-s8230" xml:space="preserve"/>
</p>
<pb file="0146" n="146"/>
<pb file="0147" n="147"/>
<pb o="1" file="0147a" n="148"/>
<figure>
<caption xml:id="echoid-caption1" xml:space="preserve">Fig: 1</caption>
</figure>
<figure>
<caption xml:id="echoid-caption2" xml:space="preserve">Fig: 2</caption>
</figure>
<figure>
<caption xml:id="echoid-caption3" xml:space="preserve">fig: 3</caption>
</figure>
<figure>
<caption xml:id="echoid-caption4" xml:space="preserve">fig: 4</caption>
</figure>
<figure>
<caption xml:id="echoid-caption5" xml:space="preserve">Fig: 5</caption>
</figure>
<figure>
<caption xml:id="echoid-caption6" xml:space="preserve">Fig: 6</caption>
</figure>
<figure>
<caption xml:id="echoid-caption7" xml:space="preserve">Fig: 7</caption>
</figure>
<figure>
<caption xml:id="echoid-caption8" xml:space="preserve">Fig: 8</caption>
</figure>
<figure>
<caption xml:id="echoid-caption9" xml:space="preserve">Fig: 9</caption>
</figure>
<figure>
<caption xml:id="echoid-caption10" xml:space="preserve">Fig: 10</caption>
</figure>
<figure>
<caption xml:id="echoid-caption11" xml:space="preserve">Fig: 11</caption>
</figure>
<figure>
<caption xml:id="echoid-caption12" xml:space="preserve">Fig: 12</caption>
</figure>
<pb file="0148" n="149"/>
<pb file="0149" n="150"/>
<pb o="2" file="0149a" n="151"/>
  <figure>
    <image file="0149a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0149a-01"/>
  </figure>
<pb file="0150" n="152"/>
<pb file="0151" n="153"/>
<pb o="3" file="0151a" n="154"/>
  <figure>
    <image file="0151a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0151a-01"/>
  </figure>
<pb file="0152" n="155"/>
<pb file="0153" n="156"/>
<pb o="4" file="0153a" n="157"/>
  <figure>
    <image file="0153a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0153a-01"/>
  </figure>
<pb file="0154" n="158"/>
<pb file="0155" n="159"/>
<pb o="5" file="0155a" n="160"/>
  <figure>
    <image file="0155a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0155a-01"/>
  </figure>
<pb file="0156" n="161"/>
<pb file="0157" n="162"/>
<pb o="6" file="0157a" n="163"/>
  <figure>
    <image file="0157a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0157a-01"/>
  </figure>
<pb file="0158" n="164"/>
<pb file="0159" n="165"/>
<pb o="7" file="0159a" n="166"/>
  <figure>
    <image file="0159a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0159a-01"/>
  </figure>
<pb file="0160" n="167"/>
<pb file="0161" n="168"/>
<pb o="8" file="0161a" n="169"/>
  <figure>
    <image file="0161a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0161a-01"/>
  </figure>
<pb file="0162" n="170"/>
<pb file="0163" n="171"/>
<pb o="9" file="0163a" n="172"/>
  <figure>
    <image file="0163a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0163a-01"/>
  </figure>
<pb file="0164" n="173"/>
<pb file="0165" n="174"/>
<pb o="10" file="0165a" n="175"/>
  <figure>
    <image file="0165a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0165a-01"/>
  </figure>
<pb file="0166" n="176"/>
<pb file="0167" n="177"/>
<pb o="11" file="0167a" n="178"/>
  <figure>
    <image file="0167a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0167a-01"/>
  </figure>
<pb file="0168" n="179"/>
<pb file="0169" n="180"/>
<pb o="12" file="0169a" n="181"/>
  <figure>
    <image file="0169a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0169a-01"/>
  </figure>
<pb file="0170" n="182"/>
<pb file="0171" n="183"/>
<pb o="13" file="0171a" n="184"/>
  <figure>
    <image file="0171a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0171a-01"/>
  </figure>
<pb file="0172" n="185"/>
<pb file="0173" n="186"/>
<pb o="14" file="0173a" n="187"/>
  <figure>
    <image file="0173a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0173a-01"/>
  </figure>
<pb file="0174" n="188"/>
<pb file="0175" n="189"/>
<pb o="15" file="0175a" n="190"/>
  <figure>
    <image file="0175a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0175a-01"/>
  </figure>
<pb file="0176" n="191"/>
<pb file="0177" n="192"/>
</div>
<div xml:id="echoid-div214" type="section" level="1" n="27">
<head xml:id="echoid-head30" xml:space="preserve"><emph style="sc">Benevolo</emph> <emph style="sc">Lectori</emph>.</head>
<p style="it">
  <s xml:id="echoid-s8231" xml:space="preserve">_E_Lectionibus bis (quas jam quo-<lb/>dammodò poſtbumas accipis) ſep-<lb/>tem, unâ ſepoſitâ, poſtremas Op-<lb/>ticis illis, quæ nuper editæ pro-<lb/>ſtant, Comitis &amp; </s>
  <s xml:id="echoid-s8232" xml:space="preserve">quaſi Mantiſſas dectinâ-<lb/>ram; </s>
  <s xml:id="echoid-s8233" xml:space="preserve">aliàs, opinor, de proferendis in apri-<lb/>cum ejuſmodi quiſquiliis nibil cogitaturus. <lb/></s>
  <s xml:id="echoid-s8234" xml:space="preserve">Sed cum nibilominùs è re ſua fore cenſeret <lb/>Librarius ab iſtis divulſas bas ſeorſum com-<lb/>parere; </s>
  <s xml:id="echoid-s8235" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s8236" xml:space="preserve">ad comparandum buic Opel-<lb/>læ ſpeciem aliquam (ut ea nempe rejecta-<lb/>nei Schediaſmatis molem tranſcenderet) <lb/>aliud quidpiam ſuppeditari cuperet; </s>
  <s xml:id="echoid-s8237" xml:space="preserve">ejus <lb/>(baud gravatìm non dixero) votis obſecun-<lb/>dans, adjeci Lectionès priores quinque; </s>
  <s xml:id="echoid-s8238" xml:space="preserve">ſub-<lb/>ſequentibus illis materiâ agnatas, &amp; </s>
  <s xml:id="echoid-s8239" xml:space="preserve">quaſz <lb/>cobærentes; </s>
  <s xml:id="echoid-s8240" xml:space="preserve">quas ſcilicet ante aliquot annos
<pb file="0178" n="193" rhead=""/>
ut nullo animo evulgandi, ità procul ab ea <lb/>cura conceperam, quæ talem animum dece-<lb/>ret; </s>
  <s xml:id="echoid-s8241" xml:space="preserve">Enimverò craſsiùs &amp; </s>
  <s xml:id="echoid-s8242" xml:space="preserve">@πολ{αι}ότε&amp;</s>
  <s xml:id="echoid-s8243" xml:space="preserve"><unsure/>ν ſcriptæ <lb/>ſunt, neque firmè quicquam continent, ex-<lb/>tra Tyronum, quibus accommodatæ ſunt, <lb/>uſum, captúmve jacens. </s>
  <s xml:id="echoid-s8244" xml:space="preserve">quapropter barum <lb/>rerum peritos obteſtor, ut ab iis prorſus ab-<lb/>ſtineant oculos, vel ut veniam ſaltem paullò <lb/>liberaliùs indulgeant. </s>
  <s xml:id="echoid-s8245" xml:space="preserve">alter as quas dixi <lb/>ſeptem conſpectui tuo lubentiùs expono, non-<lb/>nulla ſperans in illis baberi, quæ nec eruditi-<lb/>ores piguerit inſpicere. </s>
  <s xml:id="echoid-s8246" xml:space="preserve">Vltimam amicus <lb/>(vir ſanè cum primis probus, aſt in bujuſ-<lb/>modi negotiis Flagitator improbus) extor-<lb/>ſit, aut certè, pro jure quod meritò obtinet <lb/>ſuo, exegit. </s>
  <s xml:id="echoid-s8247" xml:space="preserve">Cæterùm quid tractent, &amp; </s>
  <s xml:id="echoid-s8248" xml:space="preserve"><lb/>quorſum tendant, facilè ſingularum initia <lb/>delibans edoceberis; </s>
  <s xml:id="echoid-s8249" xml:space="preserve">ut non ſit cur te longiùs <lb/>morer aut detineam.</s>
  <s xml:id="echoid-s8250" xml:space="preserve"/>
</p>
<pb file="0179" n="194"/>
</div>
<div xml:id="echoid-div215" type="section" level="1" n="28">
<head xml:id="echoid-head31" xml:space="preserve">Lectio I.</head>
<p>
  <s xml:id="echoid-s8251" xml:space="preserve">NOvum jam ingredior dicendi campum, amæniorem ſanè <lb/>neſcio vel feraciorem, uberrimâ varietate confertum, eó-<lb/>que delectabilem; </s>
  <s xml:id="echoid-s8252" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8253" xml:space="preserve">quia primas fermè _Mathematicarum_ <lb/>_hypotheſium origines_ recludit (è quibus nempe _magnitudinum_ <lb/>_cùm deſinitiones effor mantur; </s>
  <s xml:id="echoid-s8254" xml:space="preserve">tum preprietates emer gunt_) neceſſariò <lb/>perquam utilem. </s>
  <s xml:id="echoid-s8255" xml:space="preserve">De magnitudinum intelligo generatione; </s>
  <s xml:id="echoid-s8256" xml:space="preserve">ſeu de <lb/>modis, quibus ortæ productæve concipiantur variæ magnitudinum <lb/>ſpecies. </s>
  <s xml:id="echoid-s8257" xml:space="preserve">Nec ulla certè magnitudo datur, quæ non innumeris modis <lb/>&amp; </s>
  <s xml:id="echoid-s8258" xml:space="preserve">intelligi producta poſſit, &amp; </s>
  <s xml:id="echoid-s8259" xml:space="preserve">reverà produci. </s>
  <s xml:id="echoid-s8260" xml:space="preserve">Poſſunt autem, qui <lb/>ſaltem hactenus uſurpati ſunt, ad præcipua quædam genera referri, <lb/>quorum ſe mihi jam cogitanti ſuggerentia ſunt hæc; </s>
  <s xml:id="echoid-s8261" xml:space="preserve">_per motus locales;_ <lb/></s>
  <s xml:id="echoid-s8262" xml:space="preserve">_per interſectiones magnitudinum; </s>
  <s xml:id="echoid-s8263" xml:space="preserve">per quantitate poſitionéque deter-_ <lb/>_minatas ab aſſignatis locis. </s>
  <s xml:id="echoid-s8264" xml:space="preserve">diſtantias; </s>
  <s xml:id="echoid-s8265" xml:space="preserve">per ductus magnitudinum in_ <lb/>_magnitudines; </s>
  <s xml:id="echoid-s8266" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8267" xml:space="preserve">per applicationes magnitudinum ad magnitudines;_ </s>
  <s xml:id="echoid-s8268" xml:space="preserve"><lb/>_per aggregationem magnitudinum ordine certo diſpoſitarum; </s>
  <s xml:id="echoid-s8269" xml:space="preserve">per appo-_ <lb/>_ſitionem magnitudinum ad alias, vel ſubductionem ab aliis; </s>
  <s xml:id="echoid-s8270" xml:space="preserve">per orga-_ <lb/>_nicam demum_ (ab horum quocunque deductam, aut ordinatam) <lb/>_effectionem._ </s>
  <s xml:id="echoid-s8271" xml:space="preserve">Horum, &amp; </s>
  <s xml:id="echoid-s8272" xml:space="preserve">ſi qui ſunt aliorum modus primarius, &amp; </s>
  <s xml:id="echoid-s8273" xml:space="preserve">quem <lb/>alii cuncti quodammodo ſupponant oportet, utpote ſine quo nil pro-<lb/>creari poteſt, eſt iſte, _qui per motum localem._ </s>
  <s xml:id="echoid-s8274" xml:space="preserve">De quo proinde primo <lb/>diſpiciendum. </s>
  <s xml:id="echoid-s8275" xml:space="preserve">De motu celebratur illud _Ariſtotelis_ effatum, <lb/>ἁναγμ<unsure/>@@ον ἁγνο{ου}μí<unsure/>νκς ἁυ@ς (κιν@σεως) ἁγνοε<unsure/>@ ηαἱ {τὴν} φύσιν: </s>
  <s xml:id="echoid-s8276" xml:space="preserve">igno-<lb/>rato motn neceſſariò naturam ignorari; </s>
  <s xml:id="echoid-s8277" xml:space="preserve">in Phyſicis ideò paginam u-<lb/>
<anchor type="note" xlink:label="note-0179-01a" xlink:href="note-0179-01"/>
tramque facit; </s>
  <s xml:id="echoid-s8278" xml:space="preserve">nec immeritò, cùm in natura (ſaltem quantùm hu-<lb/>manus intellectus aſſequi valet, aut experientia commonſtrare) quic-<lb/>quid ſiat, à motu ſiat, aut certè non abſque motu. </s>
  <s xml:id="echoid-s8279" xml:space="preserve">De natura motûs <lb/>igitur, &amp; </s>
  <s xml:id="echoid-s8280" xml:space="preserve">rectâ deſinitione; </s>
  <s xml:id="echoid-s8281" xml:space="preserve">de cauſis, de diſſerentiis complura <lb/>ſubtiliter argutantur Phyſici, quorum ferè _Matbematicis nibil cordi_
<pb o="2" file="0180" n="195" rhead=""/>
_vel cura._ </s>
  <s xml:id="echoid-s8282" xml:space="preserve">Sufficere poteſt his quæ communis ſenſus agnoſcit, &amp; </s>
  <s xml:id="echoid-s8283" xml:space="preserve">ob-<lb/>via comprobant experimenta pro conceſſis arripere; </s>
  <s xml:id="echoid-s8284" xml:space="preserve">hoc imprimìs <lb/>generale, Quamvis magnitudinem (magnitudinibus etiam punctum <lb/>accenſebo ceu minimum magnum, ut &amp; </s>
  <s xml:id="echoid-s8285" xml:space="preserve">inſinitum ceu maximum mag-<lb/>num, quibus mediæ interjacent magnitudines omnes ſinitæ) mobi-<lb/>lem eſſe, hoc eſt eo quo conſpicimus indies ſieri modo locum ſuum <lb/>&amp; </s>
  <s xml:id="echoid-s8286" xml:space="preserve">ſitum poſſe demutare, juxta differentias præſtitutas, motu nempe <lb/>vel directo, vel circulari; </s>
  <s xml:id="echoid-s8287" xml:space="preserve">æquabiliter veloce, vel utcunque magis <lb/>accelerato, vel magìs retardato. </s>
  <s xml:id="echoid-s8288" xml:space="preserve">Hujuſmodi dico motuum quemvis <lb/>pro lubitu ſuo tanquam evidenter poſſibilem aſſumunt, ut quid exinde <lb/>conſequatur inveſtigent &amp; </s>
  <s xml:id="echoid-s8289" xml:space="preserve">oſtendant. </s>
  <s xml:id="echoid-s8290" xml:space="preserve">De iis igitur differentiis motuum <lb/>quotæ ſint &amp; </s>
  <s xml:id="echoid-s8291" xml:space="preserve">quales diſſeremus. </s>
  <s xml:id="echoid-s8292" xml:space="preserve">In motu potiſſimum à _Mathema-_ <lb/>_ticis_ conſiderantur _ipſe modus lationis, &amp; </s>
  <s xml:id="echoid-s8293" xml:space="preserve">quantitas vis motivæ_. </s>
  <s xml:id="echoid-s8294" xml:space="preserve">ipſe <lb/>modus primo lationis, juxta quem motus, alii progreſſivi ſunt, alii <lb/>circumlatitii, alii compoſiti ex his; </s>
  <s xml:id="echoid-s8295" xml:space="preserve">tum vis motivæ quantitas, prop-<lb/>ter quam alter alterius reſpectu velocior, tardior, æquè velox; </s>
  <s xml:id="echoid-s8296" xml:space="preserve">aut <lb/>in ſe æ quabilis, acceleratus, retardatus aſſirmatur. </s>
  <s xml:id="echoid-s8297" xml:space="preserve">Ex his manant <lb/>ſontibus differentiæ motuum; </s>
  <s xml:id="echoid-s8298" xml:space="preserve">quorum de poſteriore nos primùm age-<lb/>mus, quia nonnulla continet ὲξω{τε}{ει}{κο} quæ velim quam primum <lb/>ablegata, quo reliqua poſtmodùm expeditiùs fluant &amp; </s>
  <s xml:id="echoid-s8299" xml:space="preserve">limpidiùs. </s>
  <s xml:id="echoid-s8300" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8301" xml:space="preserve"><lb/>quia vis motivæ quantitas ſine tempore dignoſci nequit, de temporis <lb/>natura perſtringendum eſt aliquid. </s>
  <s xml:id="echoid-s8302" xml:space="preserve">Tempus autem dic ſodes, quid eſt? <lb/></s>
  <s xml:id="echoid-s8303" xml:space="preserve">illud _Auguſtini_ tritiſſimum noſtis; </s>
  <s xml:id="echoid-s8304" xml:space="preserve">ſi nemo quærat ſcio, ſi quis in-<lb/>terroget neſcio. </s>
  <s xml:id="echoid-s8305" xml:space="preserve">Verùm quia _Mathematici_ crebrò tempus adhibent, <lb/>quid eo deſignetur vocabulo diſtinctè concipiant oportet; </s>
  <s xml:id="echoid-s8306" xml:space="preserve">agyrtæ <lb/>ſecùs futuri. </s>
  <s xml:id="echoid-s8307" xml:space="preserve">quare jure reſponſum exigatis; </s>
  <s xml:id="echoid-s8308" xml:space="preserve">ac ſtatim pareo, ſed <lb/>breviter ac ſimpliciter, &amp; </s>
  <s xml:id="echoid-s8309" xml:space="preserve">quantùm potero λε{πι}ολογήμα{τα} defugiens. </s>
  <s xml:id="echoid-s8310" xml:space="preserve">Ab-<lb/>ſtractè loquendo, tempus eſt perſeverantia rei cujuſque in ſuo eſſe. </s>
  <s xml:id="echoid-s8311" xml:space="preserve">Ali-<lb/>as verò res aliis diutiùs in eſſe ſuo permanere; </s>
  <s xml:id="echoid-s8312" xml:space="preserve">ſuiſſe cùm hæ non erant, <lb/>eſſe cum hæ non ſunt; </s>
  <s xml:id="echoid-s8313" xml:space="preserve">priùs incepiſſe, ſeriùs deſinere; </s>
  <s xml:id="echoid-s8314" xml:space="preserve">neque non aliquas <lb/>cum aliis unà oriri ac occidere, ſimultaneóq; </s>
  <s xml:id="echoid-s8315" xml:space="preserve">quaſi durationis progreſſu, <lb/>à carceribus ad metas, univerſum ætatis curriculum emetiri, nemini <lb/>non perſpectum eſt. </s>
  <s xml:id="echoid-s8316" xml:space="preserve">Ergo tempus abſolutè quantum eſt; </s>
  <s xml:id="echoid-s8317" xml:space="preserve">ut quantitatis <lb/>admittens (modo ſuo) præcipuas affectiones æqualitatem, inæqua-<lb/>litatem, proportionem; </s>
  <s xml:id="echoid-s8318" xml:space="preserve">nec enim diffiteatur quiſquam, opinor, <lb/>ί<unsure/>σόκ{ρο}@α fore, quæ ſimul exoriuntur &amp; </s>
  <s xml:id="echoid-s8319" xml:space="preserve">ſimul intereunt; </s>
  <s xml:id="echoid-s8320" xml:space="preserve">inæqualiter <lb/>durâſſe, quorum unum fuit antequam alterum cæperit eſſè, necnon <lb/>eſſe perſeverat, poſtquam alterum deſiêrit exiſtere. </s>
  <s xml:id="echoid-s8321" xml:space="preserve">Longius autem, <lb/>&amp; </s>
  <s xml:id="echoid-s8322" xml:space="preserve">brevius tempus nemo non dicere ſolet, nemo non concipere vide-<lb/>tur. </s>
  <s xml:id="echoid-s8323" xml:space="preserve">Quantitatis igitur particeps eſſe tempus communis ſenſus agnoſcit,
<pb o="3" file="0181" n="196" rhead=""/>
pro modo permanentiæ rerum in ſuo eſſe. </s>
  <s xml:id="echoid-s8324" xml:space="preserve">At enim dices: </s>
  <s xml:id="echoid-s8325" xml:space="preserve">ante res <lb/>omnes conditas annon tempus fuit? </s>
  <s xml:id="echoid-s8326" xml:space="preserve">extra mundum, ubi nihil manet, <lb/>annon tempus labitur? </s>
  <s xml:id="echoid-s8327" xml:space="preserve">reſpondeo, ſicut ante conditum mundum <lb/>fuit ſpatium, &amp; </s>
  <s xml:id="echoid-s8328" xml:space="preserve">extra mundum nunc eſt &amp; </s>
  <s xml:id="echoid-s8329" xml:space="preserve">quidem infinitum cui <lb/>Deus coëxiſtit) quatenus<unsure/> pe<unsure/>tuerunt olim, &amp; </s>
  <s xml:id="echoid-s8330" xml:space="preserve">poſſunt jam exiſtere <lb/>talia tantáque corpora, quæ tum non fuerunt, aut jam non ſunt; <lb/></s>
  <s xml:id="echoid-s8331" xml:space="preserve">ità priùs mundo, &amp; </s>
  <s xml:id="echoid-s8332" xml:space="preserve">ſimul cum mundo (licèt extra mundum) tem-<lb/>pus fuit, &amp; </s>
  <s xml:id="echoid-s8333" xml:space="preserve">eſt, quatenus ante mundum exortum potuerunt aliquæ <lb/>res in eſſe tamdiu permanere, poſſint jam extra mundum talis per-<lb/>manentiæ capaces res exiſtere; </s>
  <s xml:id="echoid-s8334" xml:space="preserve">potuit _Sol_ multo prius in lucem emer-<lb/>ſiſſe; </s>
  <s xml:id="echoid-s8335" xml:space="preserve">poſſit jam ille, vel alius talis ſpatiis imaginariis aſſulgere. </s>
  <s xml:id="echoid-s8336" xml:space="preserve"><lb/>Tempus igitur non actualem exiſtentiam, at capacitatem tantùm ſeu <lb/>poſſibilitatem denotat permanentis exiſtentiæ; </s>
  <s xml:id="echoid-s8337" xml:space="preserve">ſicut ſpatium capacita-<lb/>tem deſignat magnitudinis intercedentis. </s>
  <s xml:id="echoid-s8338" xml:space="preserve">Sed mirum, ingeres, ſe-<lb/>cluſo motu tempus explicari; </s>
  <s xml:id="echoid-s8339" xml:space="preserve">annon tempus motum implicat? </s>
  <s xml:id="echoid-s8340" xml:space="preserve">Mi-<lb/>nimé dico quoad abſolutam, &amp; </s>
  <s xml:id="echoid-s8341" xml:space="preserve">intrinſecam naturam ſuam; </s>
  <s xml:id="echoid-s8342" xml:space="preserve">haud ma-<lb/>gìs quàm quietem; </s>
  <s xml:id="echoid-s8343" xml:space="preserve">à neutro temporis quantitas in ſe dependet; </s>
  <s xml:id="echoid-s8344" xml:space="preserve">ſeu <lb/>currant res, ſeu ſtent; </s>
  <s xml:id="echoid-s8345" xml:space="preserve">ſeu dormiamus nos, ſive vigilemus æquo <lb/>tenore tempus labitur. </s>
  <s xml:id="echoid-s8346" xml:space="preserve">Finge ſtellas omnes ab incunabulis ſuis fixas <lb/>perſtitiſſe; </s>
  <s xml:id="echoid-s8347" xml:space="preserve">nihil indè quicquam tempori deceſſiſſet; </s>
  <s xml:id="echoid-s8348" xml:space="preserve">tamdiu quies iſta <lb/>perdurâſſet, quamdiu motus hic eſſluxit. </s>
  <s xml:id="echoid-s8349" xml:space="preserve">Prius, poſterius, ſimul <lb/>(quoad ortus rerum &amp; </s>
  <s xml:id="echoid-s8350" xml:space="preserve">interitus) etiam in illo tranquillo ſtatu fuiſſet <lb/>in ſe, potuiſſet à mente magìs perfecta apprehendi. </s>
  <s xml:id="echoid-s8351" xml:space="preserve">Sed prout ipſæ <lb/>magnitudines ſunt abſolutè quantæ, independenter ab omni menſuræ <lb/>reſpectu, etſi nos ipſarum quantitates niſi menſuras applicando per-<lb/>cipere nequeamus; </s>
  <s xml:id="echoid-s8352" xml:space="preserve">ità per ſe tempus quantum eſt, etſi quo temporis <lb/>quantitas a nobis dignoſcatur, advocandum ſit motûs ſubſidium, ceu <lb/>menſuræ quâ temporum quantitates æſtimemus, &amp; </s>
  <s xml:id="echoid-s8353" xml:space="preserve">inter ſe confera-<lb/>mus; </s>
  <s xml:id="echoid-s8354" xml:space="preserve">adeóque tempus ut menſurabile motum connotat, nec enim, <lb/>ſi res omnes immotæ perſtarent, ullo pacto quantum eſſluxiſſet tem-<lb/>poris poſſemus internoſcere; </s>
  <s xml:id="echoid-s8355" xml:space="preserve">rerum ætas indiſcreta nobis, &amp; </s>
  <s xml:id="echoid-s8356" xml:space="preserve">imper-<lb/>ceptibilis cederet. </s>
  <s xml:id="echoid-s8357" xml:space="preserve">Temporis ſſuxum non perciperemus dico? </s>
  <s xml:id="echoid-s8358" xml:space="preserve">Imo nec <lb/>aliud quippiam, at ſtupore continuo defixi ceu ſtipites conſiſteremus <lb/>aut ſaxa. </s>
  <s xml:id="echoid-s8359" xml:space="preserve">Nihil enim animadvertimus niſi quatenus aliqua mutatio ſen-<lb/>ſum aſſiciens nos interpellat, aut interna mentis operatio noſtram <lb/>conſcientiam laceſſit, ac excitat. </s>
  <s xml:id="echoid-s8360" xml:space="preserve">Ex motûs forinſecùs impellentis, <lb/>aut intra nos tumultuantis extenſione, vel intenſione diverſos rerum <lb/>gradus &amp; </s>
  <s xml:id="echoid-s8361" xml:space="preserve">quantitates æſtimamus. </s>
  <s xml:id="echoid-s8362" xml:space="preserve">Ità motûs quantitas, in quantum <lb/>a nobis obſervari poteſt, à motûs extenſione dependet;</s>
  <s xml:id="echoid-s8363" xml:space="preserve"/>
</p>
<div xml:id="echoid-div215" type="float" level="2" n="1">
<note position="right" xlink:label="note-0179-01" xlink:href="note-0179-01a" xml:space="preserve">3 _Phyſ. I._</note>
</div>
<pb o="4" file="0182" n="197" rhead=""/>
<p style="it">
  <s xml:id="echoid-s8364" xml:space="preserve">Nec per ſe quenquam tempus ſentire fatendum eſt <lb/>Semotum ab rerum motu placidáque quiete;</s>
  <s xml:id="echoid-s8365" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s8366" xml:space="preserve">Haud malè dixit Lucretius. </s>
  <s xml:id="echoid-s8367" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8368" xml:space="preserve">Philoſophus ipſe; </s>
  <s xml:id="echoid-s8369" xml:space="preserve">“Oταν {γρ}<unsure/> αὺπὶ μν<unsure/>δὲν <lb/>
<anchor type="note" xlink:label="note-0182-01a" xlink:href="note-0182-01"/>
μετασ<unsure/>άλλωμεν {τὴν} διά@ωιαν, ῆλάθ<unsure/>ωμεν μεταβά <lb/>{λλ}ον{τε}ς, {ου} δοκεῖ η<unsure/>μῖν {γρ}{γρ}νέναι <lb/>{γρ}éγ{ος}. </s>
  <s xml:id="echoid-s8370" xml:space="preserve">Re<unsure/>ctè quidem hoc, non videtur nobis, non apparet à ſomno <lb/>excitatis quantum temporis interceſſit; </s>
  <s xml:id="echoid-s8371" xml:space="preserve">at non hinc rectè colligitur, <lb/>Φανε{ρὁ}γ @π {ου}λ {ἐστι}ίν ἄν{εμ} κγ νήσεως {καὶ} μεταβλῆς ὁ {χο}ὁν{ος}. </s>
  <s xml:id="echoid-s8372" xml:space="preserve">Non perſen-<lb/>tiſcimus, ergò non eſt, illatio fallax, &amp; </s>
  <s xml:id="echoid-s8373" xml:space="preserve">fallax ſomnus, qui fecit ut <lb/>nos duo ſemota temporis inſtantia connecteremus. </s>
  <s xml:id="echoid-s8374" xml:space="preserve">interim veriſſi-<lb/>mum illud; </s>
  <s xml:id="echoid-s8375" xml:space="preserve">@{σκ} η<unsure/> @ιησις, {το}σ@τ{ος} {καὶ} ό {γρ}όν{ος} ἁ@ δοκεῖ {γρ}{γρ}νέναι, <lb/>quantus nempe motus fuit, tantum tempus videtur extitiſſe; </s>
  <s xml:id="echoid-s8376" xml:space="preserve">neque <lb/>quum tantum tempus dicimus, aliud conſuevimus intelligere, quam <lb/>tantum motum intercedere potuiſſe, cujus ſcilicet extenſioni continuò <lb/>ſucceſſivæ rerum permanentiam imaginamur coëxtendi. </s>
  <s xml:id="echoid-s8377" xml:space="preserve">Cæterùm quia <lb/>tempus alveo ſemper æquali, non per vices nunc ſegniùs, tunc rapi-<lb/>diùs præterlabi concipimus (admiſsâ ſiquidem illâ diſparitate nullam <lb/>omninò computationem, aut dimenſionem admitteret) non ideò mo-<lb/>tus omnis æquè determinandæ dignoſcendæque temporis quantitati <lb/>cenſeatur accommodatus, at is præſertim qui ſummè ſimplex &amp; </s>
  <s xml:id="echoid-s8378" xml:space="preserve">uni-<lb/>formis æquabili ſemper tenore progreditur; </s>
  <s xml:id="echoid-s8379" xml:space="preserve">mobili parem ubique <lb/>vim retinente, pérque medium uniforme delato. </s>
  <s xml:id="echoid-s8380" xml:space="preserve">Quare tempori de-<lb/>terminando tale quiddam mobile deligendum eſt, quod ſaltem quoad <lb/>motûs ſui periodos æqualem conſtanter impetum ſervat; </s>
  <s xml:id="echoid-s8381" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8382" xml:space="preserve">per æqua-<lb/>le ſpatium decurrit. </s>
  <s xml:id="echoid-s8383" xml:space="preserve">Et ad communem quidem uſum accipiendus <lb/>eſt ejuſmodi motus præcipuè notabilis, in promptu cunctis obvius, &amp; </s>
  <s xml:id="echoid-s8384" xml:space="preserve"><lb/>ſenſus omnium incurrens, qualis eſt motus ſyderum, imprimìs _Solis_ <lb/>_&amp; </s>
  <s xml:id="echoid-s8385" xml:space="preserve">Lunæ_, mirificè ſibi per omnia conſtans, &amp; </s>
  <s xml:id="echoid-s8386" xml:space="preserve">orbi terrarum con-<lb/>ſpicuus; </s>
  <s xml:id="echoid-s8387" xml:space="preserve">qui proinde nedum communi gentis humanæ ſuſſragio de-<lb/>putatus, at divino Creatoris conſilio aptus natus eſt huic uſui; </s>
  <s xml:id="echoid-s8388" xml:space="preserve">à quo <lb/>nempe pronunciatum legimus: </s>
  <s xml:id="echoid-s8389" xml:space="preserve">_Fiant luminaria in ſirmamento Cæli_, <lb/>_&amp; </s>
  <s xml:id="echoid-s8390" xml:space="preserve">dividant diem ac noctem, &amp; </s>
  <s xml:id="echoid-s8391" xml:space="preserve">ſint in ſigna, &amp; </s>
  <s xml:id="echoid-s8392" xml:space="preserve">tempora<unsure/>, &amp; </s>
  <s xml:id="echoid-s8393" xml:space="preserve">dies, &amp;_</s>
  <s xml:id="echoid-s8394" xml:space="preserve"> <lb/>
<anchor type="note" xlink:label="note-0182-02a" xlink:href="note-0182-02"/>
_annos._ </s>
  <s xml:id="echoid-s8395" xml:space="preserve">At quomodò, dices, cognoſcetur _æquabili ſolem motu ferri,_ <lb/>_&amp; </s>
  <s xml:id="echoid-s8396" xml:space="preserve">unum puta diem, aut annum alteri penitus exæquari, vel aqui_ <lb/>_temporaneum eſſe?_ </s>
  <s xml:id="echoid-s8397" xml:space="preserve">Reſpondeo non aliter hoc ( excipiendo quòd à di-<lb/>vino teſtimonio colligatur) nobis innoteſcere, quàm cum aliis æqua-<lb/>libus motibus ipſum ſolis motum contendendo. </s>
  <s xml:id="echoid-s8398" xml:space="preserve">Si nempe deprehen-<lb/>datur ſolis motus in horologio ſolari (quod ſpatiorum à ſole in circulis <lb/>æquatori parallelis percurſorum penè eerto ac exquiſitè quantitates
<pb o="5" file="0183" n="198" rhead=""/>
indicat) cùm organi cujuſvis horodeictici, ſatìs accuratè conſtructi, <lb/>motibus conſentire. </s>
  <s xml:id="echoid-s8399" xml:space="preserve">Talis enim machina è fabrica ſua comparata eſt, <lb/>ſecundum motûs ſui repetitiones ſuccedaneas, æqualiter moveri; <lb/></s>
  <s xml:id="echoid-s8400" xml:space="preserve">_Clepſydram_ puta dimetiendæ diei, vel horæ deſtinatam; </s>
  <s xml:id="echoid-s8401" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8402" xml:space="preserve">quoniam <lb/>in hac aqua, vel arena quoad quantitatem ſuam, &amp; </s>
  <s xml:id="echoid-s8403" xml:space="preserve">ſiguram, vimque <lb/>deſcendendi prorſus eadem manet; </s>
  <s xml:id="echoid-s8404" xml:space="preserve">nec non vaſculum continens, &amp; </s>
  <s xml:id="echoid-s8405" xml:space="preserve"><lb/>meatus ipſam tranſmittens haud omnino variantur, tantillo ſaltem <lb/>tempore, pérque temperiem aeris conſimilem, nec ideò cauſa ſubeſt <lb/>ulla, cur non æquales in ſingulis eſſluxibus motus obire concedatur; </s>
  <s xml:id="echoid-s8406" xml:space="preserve"><lb/>ergo ſi compertum ſit, Solares motus, ſeu quoad integras periodos, <lb/>ſeu quoad partes ipſarum proportionales, organi talis repetitis mo-<lb/>tibus exquiſitè congruere, meritò pronunciandum eſt, eos prorſus <lb/>æquabiles, &amp; </s>
  <s xml:id="echoid-s8407" xml:space="preserve">uniformes fore. </s>
  <s xml:id="echoid-s8408" xml:space="preserve">Ex quo diſcurſu liquere videtur, id <lb/>quod fortè non nemini mirum videatur, cæleſtia corpora non eſſe, ex <lb/>parte rei propriéque loquendo, primarias &amp; </s>
  <s xml:id="echoid-s8409" xml:space="preserve">originales temporis <lb/>menſuras; </s>
  <s xml:id="echoid-s8410" xml:space="preserve">aſt illos potius motus, qui prope nos ſenſibus obverſantur, <lb/>&amp; </s>
  <s xml:id="echoid-s8411" xml:space="preserve">experimentis subjacent noſtris, cùm horum ope cæleſtium motuum <lb/>regularitatem dijudicemus. </s>
  <s xml:id="echoid-s8412" xml:space="preserve">Nè quidem ipſe Sol temporis idoneus ju-<lb/>dex, aut teſtis {αυ}{τί}@ς{ος} eſt, niſi quatenus horariæ machinæ ſuſſra-<lb/>gio veracitatem ſuam adteſtatur. </s>
  <s xml:id="echoid-s8413" xml:space="preserve">Nec ſanè, quod obiter interpono, <lb/>poteſt ullo pacto ſciri num periodi ſyderum ante multa ſecula tranſcur-<lb/>ſæ noſtri ſeculi revolutionibus omnino pares fuerint; </s>
  <s xml:id="echoid-s8414" xml:space="preserve">nemo ſcilicet <lb/>aſſerat certo _Methuſelam_ illum qui tantùm non mille vitæ tranſegit <lb/>annos, eo fuiſſe reverà μακροβιώ{τε}{ρο}ν qui jam ante centum annos <lb/>fato cedit. </s>
  <s xml:id="echoid-s8415" xml:space="preserve">Quid enim, ſi Sol tum junior, eóque vegetior decuplo <lb/>citiùs periodos ſuas evolverat? </s>
  <s xml:id="echoid-s8416" xml:space="preserve">Quid ſi tum aër purior, &amp; </s>
  <s xml:id="echoid-s8417" xml:space="preserve">indè cor-<lb/>porum gravitas validior eſſecerat, ut vel ipſa organa mechanica <lb/>citatiores acciperent motus, adeóque cum noſtri temporis inſtrumen-<lb/>tis comparata fidem ſuam fallerent? </s>
  <s xml:id="echoid-s8418" xml:space="preserve">_Empedocles_ quidem, apud _Plu-_ <lb/>_tarchum,_ exiſtimâſſe dicitur Solem initio dies longè prolixiores eſſe-<lb/>ciſſe. </s>
  <s xml:id="echoid-s8419" xml:space="preserve">Sed minùs id rationi conſentaneum videtur, quia tales motus{?</s>
  <s xml:id="echoid-s8420" xml:space="preserve">} <lb/>vertiginoſi ſenſim elangueſcere potiùs ſolent; </s>
  <s xml:id="echoid-s8421" xml:space="preserve">quàm invaleſcere. </s>
  <s xml:id="echoid-s8422" xml:space="preserve">Ve-<lb/>rum obiter hæc, &amp; </s>
  <s xml:id="echoid-s8423" xml:space="preserve">vix ſeriò; </s>
  <s xml:id="echoid-s8424" xml:space="preserve">revertamur in orbitam. </s>
  <s xml:id="echoid-s8425" xml:space="preserve">Temporis <lb/>(ſeu permanentiæ rerum in ſuo eſſe, ſtatu, motúve) quantitas, ut <lb/>dictum eſt, à motu quolibet dignoſcitur, bene notorio, æquabili, <lb/>(ſeu quoad partes ad hoc adh<unsure/>ibitas ſibi conſtanter æquali ac ſimili) <lb/>dein ſecundario è quibuſvis aliis motibus, qui cum illo comparati <lb/>proportione correſpondent, è cæleſtibus imprimìs, Solis potiſſimùm <lb/>ac Lunæ. </s>
  <s xml:id="echoid-s8426" xml:space="preserve">Adeò ut æqualia tempora ſint, in quibus eadem clepſydra <lb/>ſemel ac iterum, vel æquè multis viſibus exhauritur; </s>
  <s xml:id="echoid-s8427" xml:space="preserve">aut in quibus
<pb o="6" file="0184" n="199" rhead=""/>
eadem ſydera periodos eaſdem, aut ejuſdem periodi partes æquales <lb/>abſolvunt; </s>
  <s xml:id="echoid-s8428" xml:space="preserve">inæqualia verò juxta quamcunque proportionem, in <lb/>quibus limiliter, ſeu proportionaliter inæquales periodi conſumun-<lb/>tur. </s>
  <s xml:id="echoid-s8429" xml:space="preserve">Neque quiſquam objiciat tempus communiter haberi pro men-<lb/>ſura motûs, &amp; </s>
  <s xml:id="echoid-s8430" xml:space="preserve">conſequenter ad hoc motûs diſſerentias (velocioris, <lb/>tardioris, accelerati, retardati) adſumendo tempus ut præcogni-<lb/>tum definiri; </s>
  <s xml:id="echoid-s8431" xml:space="preserve">nec ideò temporis quantitatem è motu, ſed motûs <lb/>quantitatem à tempore determinari; </s>
  <s xml:id="echoid-s8432" xml:space="preserve">nil enim obſtat quo minùs tem-<lb/>pus &amp; </s>
  <s xml:id="echoid-s8433" xml:space="preserve">motus hæc ſibi mutuò præſtent oſſicia. </s>
  <s xml:id="echoid-s8434" xml:space="preserve">Sanè veluti ſpatium <lb/>ex aliqua primùm magnitudine metimur, &amp; </s>
  <s xml:id="echoid-s8435" xml:space="preserve">quantum ſit diſcimus, <lb/>è ſpatio poſteà reliquas ei congruas magnitudines æſtimanus; </s>
  <s xml:id="echoid-s8436" xml:space="preserve">ità tem-<lb/>pus primò taxamus è motu quodam, poſteà motus reliquos ex eo di-<lb/>judicamus; </s>
  <s xml:id="echoid-s8437" xml:space="preserve">quod planè nihil eſt aliud quàm mediante tempore motus <lb/>alios cum aliis comparare; </s>
  <s xml:id="echoid-s8438" xml:space="preserve">ſicut &amp; </s>
  <s xml:id="echoid-s8439" xml:space="preserve">mediante ſpatio magnitudinum <lb/>inter ſe rationes inveſtigamus. </s>
  <s xml:id="echoid-s8440" xml:space="preserve">Qui nimirum è temporum propor-<lb/>tione motuum colligit proportionem, nil aliud quam ex organorum <lb/>horologicorum, vel ex Solarium motuum ſimul decurſorum propor-<lb/>tione dictam elicit motuum rationem. </s>
  <s xml:id="echoid-s8441" xml:space="preserve">Quod certè vidit, &amp; </s>
  <s xml:id="echoid-s8442" xml:space="preserve">exertè <lb/>docuit _Ariſtoteles_: </s>
  <s xml:id="echoid-s8443" xml:space="preserve">@ μόγογ (inquit) {τὴν} κίνησιγτῶκ{ρό}νω με{τρ}{οῦ}{μεν}, α<unsure/>{λλ}α<unsure/> <lb/>{καὶ} @ῆ κινή{ει} τ<unsure/> κ{ρό}νον{δι}ὰ {τὸ} ὸ{ρι}ζε@ {ὺπ}<unsure/> ἀ{λλ}ήλων. </s>
  <s xml:id="echoid-s8444" xml:space="preserve">Porrò, quia tem-<lb/>
<anchor type="note" xlink:label="note-0184-01a" xlink:href="note-0184-01"/>
pus, ut oſtenſum, eſt quantum uniformiter extenſum, cujus omnes <lb/>partes æquabilis motûs partibus reſpectivis, ſeu ſpatiorum æquabili <lb/>motu peractorum partibus proportione reſpondent, poſſit id quàm <lb/>optimè per magnitudinem quamlibet δμοιομερῆ repræſentari, hoc eſt <lb/>menti noſtræ ſeu phantaſiæ proponi; </s>
  <s xml:id="echoid-s8445" xml:space="preserve">per ſimpliciſſimas præſertim, <lb/>quales ſunt linea recta, &amp; </s>
  <s xml:id="echoid-s8446" xml:space="preserve">circularis; </s>
  <s xml:id="echoid-s8447" xml:space="preserve">quibuſcum etiam &amp; </s>
  <s xml:id="echoid-s8448" xml:space="preserve">tempore <lb/>ſimilitudines &amp; </s>
  <s xml:id="echoid-s8449" xml:space="preserve">analogiæ non paucæ intercedunt. </s>
  <s xml:id="echoid-s8450" xml:space="preserve">Præterquam enim <lb/>quòd tempus partes habet omnino ſimilares, rationi conſentaneum <lb/>eſt ipſum velut unicâ dimenſione præditum quantum conſiderare; </s>
  <s xml:id="echoid-s8451" xml:space="preserve">ip-<lb/>ſum enim velut ex ſimplici ſupervenientium momentorum additamento, <lb/>vel ex unius momenti quaſi continuo ſluxu conſtitutum imaginamur, <lb/>&amp; </s>
  <s xml:id="echoid-s8452" xml:space="preserve">ſolam proinde longitudinem ei ſolemus attribuere; </s>
  <s xml:id="echoid-s8453" xml:space="preserve">nec ejus quan-<lb/>titatem aliàs quàm ex lineæ decurſæ longitudine determinamus. </s>
  <s xml:id="echoid-s8454" xml:space="preserve">Si-<lb/>cut, inquam, linea puncti promoti cenſetur veſtigium, à puncto <lb/>habens quò aliquatenus diviſibilis ſit, à motu verò quòd uno modo, <lb/>ſecundum longitudinem, dividi poſſit; </s>
  <s xml:id="echoid-s8455" xml:space="preserve">ità tempus velut inſtantis con-<lb/>tinuò labentis veſtigium concipiatur, ab inſtante nonnullam indiviſi-<lb/>bilitatem habens, à ſucceſſivo ſluxu quòd eatenus diſpertiri queat. </s>
  <s xml:id="echoid-s8456" xml:space="preserve">Et <lb/>ſicuti lineæ quantitas ab unica longitudine pendet motum conſe-<lb/>quente, ità temporis quantitas ab unica conſectatur velut in longum
<pb o="7" file="0185" n="200" rhead=""/>
exporrecta ſucceſſione; </s>
  <s xml:id="echoid-s8457" xml:space="preserve">quam ſpatii decurſi longitudo demonſtrat, <lb/>ac determinat. </s>
  <s xml:id="echoid-s8458" xml:space="preserve">Tempus itaque per rectam lineam ſemper deſigna-<lb/>bimus; </s>
  <s xml:id="echoid-s8459" xml:space="preserve">arbitrariè quidem initio ſumptam &amp; </s>
  <s xml:id="echoid-s8460" xml:space="preserve">expoſitam, at cujus <lb/>partes proportionalibus temporis partibus, &amp; </s>
  <s xml:id="echoid-s8461" xml:space="preserve">puncta temporis inſtan-<lb/>tibus reſpectivis juſtè reſpondebunt, &amp; </s>
  <s xml:id="echoid-s8462" xml:space="preserve">iis appoſitè repræſentandis <lb/>inſervient. </s>
  <s xml:id="echoid-s8463" xml:space="preserve">His de tempore prælibatis ad conſiderandam vim motûs <lb/>eſſectivam procedimus, quæ ſanè (quæcunque ſit ejus natura, vel <lb/>undicunque procedat, nam iſta _Phyſicis_ diſquirenda relinquimus) <lb/>merito quoque ſeu quantum quid concipitur, &amp; </s>
  <s xml:id="echoid-s8464" xml:space="preserve">ſicut alia quanta com-<lb/>puto ſubjicitur. </s>
  <s xml:id="echoid-s8465" xml:space="preserve">Etenim experientiâ compertiſſi@@m eſt, ſæpe duo-<lb/>rum mobilium ab eodem termino per eandem orbitam delatorum al-<lb/>terum alteri prævertere, ſeu majus eodem tempore ſpatium conſi-<lb/>cere. </s>
  <s xml:id="echoid-s8466" xml:space="preserve">Nec aliunde poteſt hoc procedere, quàm à majori vi, ſeu <lb/>potentia motiva, quâ præcellit alterum mobile, cujú@que gratiâ velo-<lb/>cius dicitur. </s>
  <s xml:id="echoid-s8467" xml:space="preserve">Et quia perſpicuum eſt nil impedire, quin ſecundum <lb/>omnimodas proportiones contingat hic ſpatiorum una peractorum <lb/>exceſſus, ideò vis hæc jure concipiatur in partes quaſlibet (quas &amp; </s>
  <s xml:id="echoid-s8468" xml:space="preserve"><lb/>ſicuti partes cujuſcunque qualitatis intenſivas ſuccinctæ diſtinctionis <lb/>ergò gradus appellare licet, &amp; </s>
  <s xml:id="echoid-s8469" xml:space="preserve">conſuetum eſt) in partes, inquam, <lb/>quaſlibet infinitas, aut indefinitas diviſibilis concipiatur; </s>
  <s xml:id="echoid-s8470" xml:space="preserve">quas inter <lb/>ſe nectens, &amp; </s>
  <s xml:id="echoid-s8471" xml:space="preserve">à ſe dirimens communis terminus, vel (juxta ſuppo-<lb/>ſitionem quòd quanta conſtant ex infinitis atomis) pars abſolutè mi-<lb/>nima dicatur quies, hoc eſt ſumma tarditas, aut infima velocitas; <lb/></s>
  <s xml:id="echoid-s8472" xml:space="preserve">è cujus ſuccreſcentia, vel intenſione continua velocitatis gradus quili-<lb/>bet eo modo concipiatur aggregari, vel produci, quo linea è puncto-<lb/>rum appoſitione, vel motu, tempus ex inſtantium ſucceſſione vel ſluxu <lb/>progenitum imaginamur. </s>
  <s xml:id="echoid-s8473" xml:space="preserve">Unde rem abſolutè conſiderando, quo <lb/>vis hujuſce quantitas menti ſeu phantaſiæ rectè proponatur, ſuſſicit <lb/>ejus vice magnitudinem quamvis regularem exhibere (hoc eſt talem, <lb/>in cujus partibus quamvis diſſerentiam, quamlibétque proportionem <lb/>clarè promptéque valeamus apprehendere) ſimplicitatis adeò perſpi-<lb/>cuitatíſque causâ cuilibet ejus repræſentando gradui recta linea cum <lb/>primìs accuratè quadrat. </s>
  <s xml:id="echoid-s8474" xml:space="preserve">Ità quidem in ſe generatim &amp; </s>
  <s xml:id="echoid-s8475" xml:space="preserve">abſoluta <lb/>ſpectata vis iſta tempus non implicat, eóque ſecluſo concipi poteſt (in <lb/>quolibet enim temporis inſtanti, pérque quodcunque temporis inter-<lb/>vallum eâ præditum mobile concipiatur) at quatenus computabilis, <lb/>ac æſtimio Mathematico ſubdita, quâ ratione velocitas dicitur, cum <lb/>ſpatio tempus adſignificat; </s>
  <s xml:id="echoid-s8476" xml:space="preserve">è quibus nempe quantitas ejus dijudicatur, <lb/>ac diſcernitur definitur idcircò velocitas potentia, quâ mobile ſpatium <lb/>aliquod in aliquo tempore pertranſire poteſt. </s>
  <s xml:id="echoid-s8477" xml:space="preserve">Uude conſectatur ſin-
<pb o="8" file="0186" n="201" rhead=""/>
gularem velocitatis cujuſpiam quantitatem nec ex ſola confecti ſpatii, nec <lb/>ex abſumpti temporis quantitate dignoſci poſle (quælibet enim velocitas <lb/>aliquo tempore quodvis aſſignatum ſpatium emetiatur) aſt ex ſpatii <lb/>ſimul ac temporis quantitatibus ad calculum redactis eam innoteſcere; <lb/></s>
  <s xml:id="echoid-s8478" xml:space="preserve">ſicut &amp; </s>
  <s xml:id="echoid-s8479" xml:space="preserve">viciſſim temporis abſumpti quantitas non niſi ſpatii ſimul ac <lb/>velocitatis agnitis quantitatibus determinetur. </s>
  <s xml:id="echoid-s8480" xml:space="preserve">Quinimo ſpati quo-<lb/>que quantitas (quatenus hoc modo per motum dignoſcibilis eſt) nec <lb/>è ſola definitæ velocitatis quantitate, nec ab aſſignato tanto tempore <lb/>dependet, aſt ab utriuſque ratione conjuncta. </s>
  <s xml:id="echoid-s8481" xml:space="preserve">Et quidem ut hæc quo-<lb/>modo @e reſpiciant amplius exponamus, ſpatii quatenus hoc modo <lb/>computatur quantitas eo ferè dignoſcitur modo, quo è dimenſionibus <lb/>ſuis quanta ſit ſuperficies innoteſcit; </s>
  <s xml:id="echoid-s8482" xml:space="preserve">è quantitate ſcilicet unius lineæ, <lb/>(quæ longitudinem ejus aut altitudinem oſtentat) &amp; </s>
  <s xml:id="echoid-s8483" xml:space="preserve">è quantitatibus <lb/>ſingularum invicem ſibi parallelarum linearum, quæ per iſtius lineæ <lb/>puncta quæque tranſeuntes ſuperficiem totam quodammodo conſti-<lb/>tuunt, &amp; </s>
  <s xml:id="echoid-s8484" xml:space="preserve">componunt; </s>
  <s xml:id="echoid-s8485" xml:space="preserve">eam ſa@tem limitant atque determinant; </s>
  <s xml:id="echoid-s8486" xml:space="preserve">hoc <lb/>eſt quaſi per ductum ſingularum ejuſmodi linearum in reſpectiva <lb/>dictæ lineæ puncta. </s>
  <s xml:id="echoid-s8487" xml:space="preserve">Velocitatis autem, &amp; </s>
  <s xml:id="echoid-s8488" xml:space="preserve">temporis quantitates <lb/>pariter eo modo diſcernuntur, quo ex ſuperficiei, &amp; </s>
  <s xml:id="echoid-s8489" xml:space="preserve">unius cui appli-<lb/>catur dimenſionis quantitate diſcernitur quanta ſit reliqua dimenſio <lb/>(ubivis, inquam, aut ſaltem alicubi quanta, nam fieri poteſt ut re-<lb/>liqua dimenſio quatenus per omnia prioris dimenſionis puncta diſſundi-<lb/>tur, ſibi paſſim diſpar &amp; </s>
  <s xml:id="echoid-s8490" xml:space="preserve">diſſormis ſit; </s>
  <s xml:id="echoid-s8491" xml:space="preserve">quid velim è veſtigio conſtabit, <lb/>nam utilis hæc conſideratio poſtulat enucleatiùs declarari. </s>
  <s xml:id="echoid-s8492" xml:space="preserve">Omni <lb/>temporis inſtanti, ſeu indefinitè parvæ temporis particulæ (inſtanti <lb/>dico, vel indefinitæ particulæ, nam utì nihil admodum refert, utrum <lb/>lineam ex innumeris punctis, an ex indefinitè parvis lineolis compo-<lb/>ſitam intelligamus, ita perinde eſt, utrum tempus ex inſtantibus, <lb/>an ex innumeris minutis tempuſculis conſlatum ſupponamus; </s>
  <s xml:id="echoid-s8493" xml:space="preserve">nos ſal-<lb/>tem brevitati conſulentes pro temporibus quantumlibet exiguis in-<lb/>ſtantia, hoc eſt pro tempuſcula repræſentantibus lineolis puncta non <lb/>verebimur uſurpare) cuilibet dico temporis momento competit velo-<lb/>citatis aliquis gradus, quem mobile tunc habere concipiendum eſt; </s>
  <s xml:id="echoid-s8494" xml:space="preserve"><lb/>cui gradui reſpondet aliqua decurſi ſpatii longitudo (nam hìc mobile <lb/>tanquam punctum, &amp; </s>
  <s xml:id="echoid-s8495" xml:space="preserve">ſpa tium proinde tantummodò ceu longum <lb/>conſideramus) quia veròtemporis momenta quoad rem ipſam neuti-<lb/>quam à ſe dependent, ſupponi poterit in proximo inſtanti mobile <lb/>gradum velocitatis alium (alium inquam vel æqualem priori, vel in <lb/>quavis proportione diverſum) admittere, cui proinde reſpondebit <lb/>alia ſpatii longitudo, tali proportione reſpiciens priorem, quali
<pb o="9" file="0187" n="202" rhead=""/>
velocitatis hic gradus præcedentem. </s>
  <s xml:id="echoid-s8496" xml:space="preserve">Quum enim temporis inſtantia <lb/>prorſus æqualia ſint inter ſe, ſpatialium longitudinum ratio à ſola <lb/>velocitatem ratione dependebit, eíque proinde par erit, aut ſimilis <lb/>(quod niſi pro veriſſimo ſumatur, haud ullo modo menſurari poſſit <lb/>velocitas; </s>
  <s xml:id="echoid-s8497" xml:space="preserve">nam à ſola ſpatiorum eodem tempore decurſorum (vel <lb/>eodem inſtanti) proportione velocitatum inter ſe collatarum imme-<lb/>diatè vel mediatè ratio taxatur, &amp; </s>
  <s xml:id="echoid-s8498" xml:space="preserve">altera alterius reſpectu denomi-<lb/>natur tanta) ſimiliter ſi per omnia temporis cujuſvis momenta qui <lb/>conveniunt ipſis velocitatis gradus aſſignentur, aggregabitur ex iis <lb/>quantum quiddam, cujus partibus quibuſvis decurſorum ſpatiorum <lb/>partes reſpectivæ, hoc eſt iiſdem temporibus reſpondentes particulæ, <lb/>juſtè proportionantur, adeóque quantum è gradibus iſtis conſtans <lb/>repræſentans magnitudo ſpatium quoque decurſum repræſentare poſſit; <lb/></s>
  <s xml:id="echoid-s8499" xml:space="preserve">quatenus nempe qualem ſpatii partes temporibus ſingulis peractæ pro-<lb/>portionem inter ſe ſervant, exactè referat. </s>
  <s xml:id="echoid-s8500" xml:space="preserve">Quum igitur, utpote <lb/>quàm æquabiliſſimè ſluens per lineam, ut præmonuimus, rectam ap-<lb/>tiſſimè repræſentetur, &amp; </s>
  <s xml:id="echoid-s8501" xml:space="preserve">qui in ſingulis temporis inſtantibus haben-<lb/>tur alii ac alii, ſibimet æquales; </s>
  <s xml:id="echoid-s8502" xml:space="preserve">aut inæquales, velocitatis gradus per <lb/>lineas itidem, ut priùs etiam inſinuatum eſt, rectas exprimantur, <lb/>&amp; </s>
  <s xml:id="echoid-s8503" xml:space="preserve">cùm hi velocitatis gradus ſingula temporis momenta alii ac alii <lb/>permeent, independentèr à ſe invicem ac impermixtè; </s>
  <s xml:id="echoid-s8504" xml:space="preserve">itaque ſi per <lb/>lineæ tempus repræſentantis omnia puncta trajiciantur rectæ ſic <lb/>diſpoſitæ, ut altera nulla nulli alteri coïncidat, hoc eſt in ſitu pa-<lb/>rallelo; </s>
  <s xml:id="echoid-s8505" xml:space="preserve">quæ reſultat hinc ſuperficies plana (pro quantitate temporis, <lb/>&amp; </s>
  <s xml:id="echoid-s8506" xml:space="preserve">poſitorum velocitatis graduum ratione determinata) graduum ve-<lb/>locitatis aggregatum exactiſſimè referet; </s>
  <s xml:id="echoid-s8507" xml:space="preserve">cujus ſuperficiei partes cùm <lb/>reſpectivis (ut prædictum) ſpatii peracti partibus proportionales <lb/>ſint, poterit id ſpatio quoque repræſentando commodiſſimè adaptari. </s>
  <s xml:id="echoid-s8508" xml:space="preserve"><lb/>Iſta verò ſuperficies brevitatis causâ dehinc appellabitur velocitas ag-<lb/>gregata, vel ſpatii repræſentativa. </s>
  <s xml:id="echoid-s8509" xml:space="preserve">Neque quenquam aſſiciat, nam <lb/>ſubmovenda nobis hæc remora, quod diximus in ſingulis temporis <lb/>inſtantibus longitudinem aliquam confici, quaſi dari poſſe motum <lb/>inſtantaneum aſſirmarem. </s>
  <s xml:id="echoid-s8510" xml:space="preserve">Nam poſito tempora è momentis com-<lb/>poni, etiam lineæ componentur è punctis; </s>
  <s xml:id="echoid-s8511" xml:space="preserve">quòd ſi lineæ inæ-<lb/>quales componantur è punctis infinitis, ſibimet æquinume-<lb/>ris, neceſſariò ſequitur linearum puncta, juxta ſimilem cum ipſis <lb/>proportionem inæqualia fore, adeóque per longitudines in æquitem-<lb/>poraneìs momentis decurſas duntaxat intelligenda ſunt ejuſmodi inæ-<lb/>qualia puncta, è quibus tota decurſa longitudo quaſi conſlatur. </s>
  <s xml:id="echoid-s8512" xml:space="preserve">Sin <lb/>hoc abſonum cuipiam videatur, &amp; </s>
  <s xml:id="echoid-s8513" xml:space="preserve">nullo ſenſu motus admittatur in-
<pb o="10" file="0188" n="203" rhead=""/>
ſtantaneus, eò recurrendum ut per inſtantias nil aliud, quàm inde-<lb/>finitas temporis particulas intelligamus; </s>
  <s xml:id="echoid-s8514" xml:space="preserve">quibus reſpondeant certo <lb/>velocitatis gradu, alio atque alio, percurſa indefinitè minuta ſpatiola <lb/>velocitatis gradibus adproportionata; </s>
  <s xml:id="echoid-s8515" xml:space="preserve">tum autem repræſentando <lb/>ſingulo cuipiam velocitatis gradui per tempuſculum aliquod retento, <lb/>loco lineæ rectæ ſubſtituatur oportet exiguum rectangulum dicto tem-<lb/>puſculo applicatum. </s>
  <s xml:id="echoid-s8516" xml:space="preserve">Perinde fuerit, ac eodem recidet hoc an illo <lb/>modo ſe res habeat, aſt ſimplicior &amp; </s>
  <s xml:id="echoid-s8517" xml:space="preserve">clarior videtur iſte modus, quem <lb/>priùs expoſuimus, cui proinde poſthac inſiſtemus. </s>
  <s xml:id="echoid-s8518" xml:space="preserve">Ut redeam, &amp; </s>
  <s xml:id="echoid-s8519" xml:space="preserve"><lb/>recolligam; </s>
  <s xml:id="echoid-s8520" xml:space="preserve">ſicuti per omnia lineæ rectæ puncta traduci poſſunt pa-<lb/>rallelæ rectæ, magnitudine pro lubitu pares, vel impares, è qui-<lb/>bus aggregatis ſuperficiale planum exurgat, ità ad ſingula temporis <lb/>inſtantia applicari poſſunt velocitatis gradus diverſi, pares vel impares, <lb/>prout mobile per totam ſuam lationem vel eundem impetum retmere, <lb/>vel aliquando varium adſciſſere ſupponatur, utcunque creſcendo vel <lb/>decreſcendo. </s>
  <s xml:id="echoid-s8521" xml:space="preserve">Si velocitatem ſemper eandem conſervare dicatur, fa-<lb/>cilè patet è dictis velocitatem aggregatam definito cuivis tempori con-<lb/>venientem rectiſſimè per figuram parallelogrammam exprimi, qua-<lb/>
<anchor type="note" xlink:label="note-0188-01a" xlink:href="note-0188-01"/>
lis eſt AZZE, in qua latus AE temporis deſiniti vicem obit, re-<lb/>liquum AZ, eíque parallelæ rectæ omnes BZ, CZ, DZ, EZ <lb/>velocitatis gradus ſingulos per ſingula temporis momenta penetrantes, <lb/>in hoc ſcilicet caſu pares, exhibent. </s>
  <s xml:id="echoid-s8522" xml:space="preserve">Poſſunt etiam, ut dictum, pa-<lb/>rallelogramma AZZB, AZZC, AZZD, AZZE ſpatia re-<lb/>ſpectivis temporibus AB, AC, AD, AE decurſa appoſitè deſignare. <lb/></s>
  <s xml:id="echoid-s8523" xml:space="preserve">E qua conſideratione ſola, vel intuitu primo motûs hujuſmodi, quem <lb/>æquabilem, &amp; </s>
  <s xml:id="echoid-s8524" xml:space="preserve">uniformem vocitant, omnia ſymptomata deduci <lb/>poſſunt. </s>
  <s xml:id="echoid-s8525" xml:space="preserve">Quales ſunt: </s>
  <s xml:id="echoid-s8526" xml:space="preserve">quòd æquali perpetuò velocitate tranſmiſſa <lb/>ſpatia ſeſe habent ut tempora: </s>
  <s xml:id="echoid-s8527" xml:space="preserve">Quod æquali tempore peracta <lb/>ſpatia ſeſe habent ut velocitates; </s>
  <s xml:id="echoid-s8528" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8529" xml:space="preserve">viciſſim: </s>
  <s xml:id="echoid-s8530" xml:space="preserve">Si ſpatia ſunt ut <lb/>velocitates tempora fore æqualia; </s>
  <s xml:id="echoid-s8531" xml:space="preserve">ſi ut tempora, velocitates <lb/>æquari. </s>
  <s xml:id="echoid-s8532" xml:space="preserve">Et ſi æqualia ſpatia fuerint, tempora velocitatibus propor-<lb/>tione reciprocari; </s>
  <s xml:id="echoid-s8533" xml:space="preserve">contráque, ſi tempora velocitatibus proportione <lb/>reciprocentur, ſpatia ſibimet exæquari. </s>
  <s xml:id="echoid-s8534" xml:space="preserve">Spatia denique quælibet <lb/>compoſitam habere rationem è rationibus velocitatum &amp; </s>
  <s xml:id="echoid-s8535" xml:space="preserve">temporum; </s>
  <s xml:id="echoid-s8536" xml:space="preserve"><lb/>nec non, ſubducendo rationem temporum è ratione ſpatiorum reſiduam <lb/>manere rationem velocitatum; </s>
  <s xml:id="echoid-s8537" xml:space="preserve">vel ſubducendo rationem velocitatum <lb/>relinqui rationem temporum. </s>
  <s xml:id="echoid-s8538" xml:space="preserve">Hæc enim parallelogrammorum inter <lb/>ſe comparatorum aſſectiones ſunt (æquiangulorum intelligo paralle-<lb/>logrammorum; </s>
  <s xml:id="echoid-s8539" xml:space="preserve">nam ubi repræſentativa, hæc parallelogramma con-<lb/>feruntur inter ſe, æquiangula conſtituantur oportet; </s>
  <s xml:id="echoid-s8540" xml:space="preserve">alioqui cùm
<pb o="11" file="0189" n="204" rhead=""/>
ſingillatim ſpectantur; </s>
  <s xml:id="echoid-s8541" xml:space="preserve">nihil refert quinam angulus ſtatuatur) hæc, <lb/>inquam, è parallelogrammorum natura liquent, &amp; </s>
  <s xml:id="echoid-s8542" xml:space="preserve">ex iis quæ <lb/>poſuimus ſponte conſectantur; </s>
  <s xml:id="echoid-s8543" xml:space="preserve">ut nullam aliam demonſtrationem re-<lb/>quirere videantur. </s>
  <s xml:id="echoid-s8544" xml:space="preserve">Et ſanè quoad omnes Mathematicæ {ακ}<unsure/>@ψΗ<unsure/> ſubditas <lb/>(hoc eſt utcunque quantitatem involventes) materias cùm magnâ fa-<lb/>cilitate Theoremata perſpicere, tum ſummo eadem compendio de-<lb/>monſtrare poterit, quiſquis contemplationi ſuæ ſubjecta cujuſcunque <lb/>generis quanta ad analogicas magnitudines ritè congruéque novit re-<lb/>digere. </s>
  <s xml:id="echoid-s8545" xml:space="preserve">Quòd ſi porrò velocitatis gradus continuò per ſingula tempo-<lb/>ris inſtantia ſupponantur æqualiter adaugeri, vel imminui, à gradu <lb/>minimo, ſeu quiete, definitum ad velocitatis gradum, vel à definito <lb/>tali gradu ad quietem; </s>
  <s xml:id="echoid-s8546" xml:space="preserve">conſimili pacto poterit aggregata velocitas per <lb/>quamvis ſuperficiem æqualiter à puncto creſcentem ad definitam mag-<lb/>nitudine lineam; </s>
  <s xml:id="echoid-s8547" xml:space="preserve">vel eodem retrogradè paſſu decreſcentem, exhiberi; <lb/></s>
  <s xml:id="echoid-s8548" xml:space="preserve">ſimpliciſſimè verò, &amp; </s>
  <s xml:id="echoid-s8549" xml:space="preserve">optimè per triangulum rectilineum; </s>
  <s xml:id="echoid-s8550" xml:space="preserve">ut puta per <lb/>triangulum AEY, in quo crus AE tempus denotat; </s>
  <s xml:id="echoid-s8551" xml:space="preserve">ejúſque punctis <lb/>
<anchor type="note" xlink:label="note-0189-01a" xlink:href="note-0189-01"/>
applicatæ lineæ parallelæ BY, CY, DY, EY gradus velocitatis ſin-<lb/>gulis inſtantibus congruos à puncto A (quod quietem, vel infimam <lb/>velocitatem refert) ad definitum gradum lineâ maximâ EY repræſen-<lb/>tatum æqualiter increſcentes; </s>
  <s xml:id="echoid-s8552" xml:space="preserve">vel ab eadem EY retrò ad punctum A <lb/>quietis repræſentativum declinantes. </s>
  <s xml:id="echoid-s8553" xml:space="preserve">Sed &amp; </s>
  <s xml:id="echoid-s8554" xml:space="preserve">pari jure, quo priùs, <lb/>trigona ABY, ACY, ADY, AEY per reſpectiva ab initio tem-<lb/>pora decurſis ſpatiis repræſentandis inſervient. </s>
  <s xml:id="echoid-s8555" xml:space="preserve">Et conſequenter, ſi <lb/>velocitas æqualiter à definito gradu ad gradum definitum ſupponatur <lb/>augeri, vel diminui, repræſentabitur tam aggregata velocitas, quàm <lb/>ſpatium ei reſpondens à figura quadrangula Trapezia, qualis eſt CYYE, <lb/>in ſigura priùs adhibita. </s>
  <s xml:id="echoid-s8556" xml:space="preserve">Hinc, non ſecùs quàm in præcedentibus, hu-<lb/>juſmodi motûs quem uniformiter acceleratum nomine perquam apto <lb/>_Galilæus_ nuncupavit) aſſectiones omnes præcipuæ facilimè deprehen-<lb/>dentur, atque demonſtrabuntur; </s>
  <s xml:id="echoid-s8557" xml:space="preserve">cujuſmodi ſunt:</s>
  <s xml:id="echoid-s8558" xml:space="preserve">Quòd æ quali tem-<lb/>pore conficietur æquale ſpatium per motum à quiete uniformiter acce-<lb/>leratum, ac per ipſum motum uniformem, modò velocitas hujus ſub-<lb/>dupla ſit velocitatis, quam ille maximam habet. </s>
  <s xml:id="echoid-s8559" xml:space="preserve">Quòd ſpatia motu <lb/>à quiete uniformiter accelerato peracta, ſeſe habent ut _Quadr @ta tem-_ <lb/>_porum_ (vel in duplicata temporum proportione.) </s>
  <s xml:id="echoid-s8560" xml:space="preserve">Et diverſos hoc <lb/>modo acceleratos motus comparando: </s>
  <s xml:id="echoid-s8561" xml:space="preserve">Quòd ab illis tranſacta ſpatia <lb/>habeant rationem è rationibus temporum, &amp; </s>
  <s xml:id="echoid-s8562" xml:space="preserve">velocitatum maxima-<lb/>rum: </s>
  <s xml:id="echoid-s8563" xml:space="preserve">Et ſimilia talia vel his connexa, vel indè conſequentia, quæ <lb/>triangulis conveniunt inter ſe quoad ſuas, &amp; </s>
  <s xml:id="echoid-s8564" xml:space="preserve">quoad laterum rationes <lb/>comparatis; </s>
  <s xml:id="echoid-s8565" xml:space="preserve">quæ ex poſitis haud diſſicilè perſpìciantur, ac demon-
<pb o="12" file="0190" n="205" rhead=""/>
ſtrentur. </s>
  <s xml:id="echoid-s8566" xml:space="preserve">Porrò, non abſimiliter ſi velocitatis gradus continuâ per <lb/>ſingula temporis inſtantia ſucceſſione, à quiete ad definitum gradum, <lb/>vel retrogradè, creſcere concipiantur, aut decreſcere juxta progreſſi-<lb/>onem numerorum quadraticorum repræſentatur tum optimè velocitas <lb/>
<anchor type="note" xlink:label="note-0190-01a" xlink:href="note-0190-01"/>
aggregata, ſicut &amp; </s>
  <s xml:id="echoid-s8567" xml:space="preserve">ſpatium hujuſmodi motu confectum, à comple-<lb/>mento Semiparabolæ, qualis eſt AEX, cujus vertex A quietem (ſeu <lb/>motûs ac temporis initium) tangens AE tempus definitum, linea BX <lb/>primum velocitatis accreſcentis gradum (qui ſe habet ut I.) </s>
  <s xml:id="echoid-s8568" xml:space="preserve">proxima <lb/>CX ſecundum gradum (habentem ſe ut 4.) </s>
  <s xml:id="echoid-s8569" xml:space="preserve">ſubſequens DX (qui ſe <lb/>habet ut 9.) </s>
  <s xml:id="echoid-s8570" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8571" xml:space="preserve">ità porrò uſque ad ultimum EX: </s>
  <s xml:id="echoid-s8572" xml:space="preserve">Id quod ex notiſſi-<lb/>ma parabolæ proprietate manifeſtum eſt. </s>
  <s xml:id="echoid-s8573" xml:space="preserve">Eodem planè modo quivis <lb/>ſuppoſiti velocitatis gradus, utcunque creſcentis aut decreſcentis, <lb/>continuo vel interruptè, quovis, inquam, imaginabili modo per <lb/>lineas rectas ad temporis repræſentatricem rectam applicatas certiſſi-<lb/>mo, commodiſſimòque modo deſignari poſſunt, aſſervatâ quam quis <lb/>adſignare voluerit proportione; </s>
  <s xml:id="echoid-s8574" xml:space="preserve">ſic ut inde cognitâ ſpatii repræſen-<lb/>tantis dimenſione, ſpatii per motum confecti quantitas faciliùs inno-<lb/>teſcat; </s>
  <s xml:id="echoid-s8575" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8576" xml:space="preserve">reciprocè, cognitâ ſpatii dicti naturâ velocitatis ac tem-<lb/>poris quantitatibus dignoſcendis aliqua lux aſſulgeat: </s>
  <s xml:id="echoid-s8577" xml:space="preserve">Quæ quidem <lb/>poſthac dicendorum intellectui neceſſaria, totíque motuum theoriæ <lb/>non parùm ut videtur utilia viſum eſt paullo fuſiùs expoſita præmittere. <lb/></s>
  <s xml:id="echoid-s8578" xml:space="preserve">Quà perfunctus operâ pedem figo.</s>
  <s xml:id="echoid-s8579" xml:space="preserve"/>
</p>
<div xml:id="echoid-div216" type="float" level="2" n="2">
<note position="left" xlink:label="note-0182-01" xlink:href="note-0182-01a" xml:space="preserve">_Phyſ. IV._ 16.</note>
<note position="left" xlink:label="note-0182-02" xlink:href="note-0182-02a" xml:space="preserve"><gap/> I. 14.</note>
<note position="left" xlink:label="note-0184-01" xlink:href="note-0184-01a" xml:space="preserve">_Phy ſ. IV. 18._</note>
<note position="left" xlink:label="note-0188-01" xlink:href="note-0188-01a" xml:space="preserve">Fig. I.</note>
<note position="right" xlink:label="note-0189-01" xlink:href="note-0189-01a" xml:space="preserve">Fig. I.</note>
<note position="left" xlink:label="note-0190-01" xlink:href="note-0190-01a" xml:space="preserve">Fig. 2.</note>
</div>
<pb o="13" file="0191" n="206"/>
</div>
<div xml:id="echoid-div218" type="section" level="1" n="29">
<head xml:id="echoid-head32" xml:space="preserve"><emph style="sc">Lect</emph>. II.</head>
<p>
  <s xml:id="echoid-s8580" xml:space="preserve">VArios, quibus productæ concipiantur magnitudines aggreſſi mo-<lb/>dos conſiderare, primum &amp; </s>
  <s xml:id="echoid-s8581" xml:space="preserve">præcipuum attingere cæpimus illum, <lb/>qui motu peragitur locali. </s>
  <s xml:id="echoid-s8582" xml:space="preserve">Cùm verò ſoleant _Matbematici_ diverſi-<lb/>modos, è quibus aliæ ac aliæ magnitudines reſultant, motus adſumere <lb/>ceu poſſibiles, duos ad fontes digitum intendimus, è quibus iſtæ mo-<lb/>tuum differentiæ ſcaturiunt, modum lationis ipſum, &amp; </s>
  <s xml:id="echoid-s8583" xml:space="preserve">quantita-<lb/>tem vis motivæ; </s>
  <s xml:id="echoid-s8584" xml:space="preserve">quorum poſteriopem haud ita clarum &amp; </s>
  <s xml:id="echoid-s8585" xml:space="preserve">apertum <lb/>nuperrimè conati ſumus recludere, limpidúmque reddere. </s>
  <s xml:id="echoid-s8586" xml:space="preserve">Jam diffe-<lb/>rentias quas aſſumunt ipſas proſequemur, &amp; </s>
  <s xml:id="echoid-s8587" xml:space="preserve">quo pacto generationi mag-<lb/>nitudinum inſervire poſſunt oſtendemus. </s>
  <s xml:id="echoid-s8588" xml:space="preserve">Lationis modum ſpectando <lb/>generantur magnitudines vel per motus ſimplices, vel per motus com-<lb/>poſitos, vel ex concurſu motuum (nam compoſitionem à concurſu <lb/>diſtinguo, quæ tamen à nonnullis confunduntur.) </s>
  <s xml:id="echoid-s8589" xml:space="preserve">De ſimplicium <lb/>motuum hypotheſibus, ac effectis primò videamus. </s>
  <s xml:id="echoid-s8590" xml:space="preserve">Simplicium mo-<lb/>tuum duo genera ſunt, ρο@, &amp; </s>
  <s xml:id="echoid-s8591" xml:space="preserve">{πο}<unsure/>{ρι}φο@, progreſſio, &amp; </s>
  <s xml:id="echoid-s8592" xml:space="preserve">circum-<lb/>latio. </s>
  <s xml:id="echoid-s8593" xml:space="preserve">Sub progreſſivo motu comprehenditur motus omnis, qui nul-<lb/>lum fixum locum (loci nomine quamvis magnitudinem, etiam pun-<lb/>ctum adnumerans, intelligo,) reſpicit, cui velut innectitur, ac affigi-<lb/>tur; </s>
  <s xml:id="echoid-s8594" xml:space="preserve">ſeu directus iſte motus ſit, ſeu reflexus, ſeu refractus; </s>
  <s xml:id="echoid-s8595" xml:space="preserve">ſive <lb/>callem certum perſequatur, ſive inconſtanter deſultet, divagetur, <lb/>exorbitet. </s>
  <s xml:id="echoid-s8596" xml:space="preserve">Quia vero penitus irregularium in arte nulla ratio poteſt <lb/>haberi, ſufficit _Matbematicis_ ſupponere magnitudinem quamcunque <lb/>progredi poſſe juxta deſignatam quamlibet orbitam; </s>
  <s xml:id="echoid-s8597" xml:space="preserve">ut _v.</s>
  <s xml:id="echoid-s8598" xml:space="preserve">g._ </s>
  <s xml:id="echoid-s8599" xml:space="preserve">Quod <lb/>punctum _in linea recta, circulari, elliptica, ſpirali, vel alia quavis_ <lb/>_præſtituta queat incedere._ </s>
  <s xml:id="echoid-s8600" xml:space="preserve">Verùm præcipuæ, hoc eſt maximi, fre-<lb/>quentiſſimíque pro magnitudinibus efformandis usûs, circa hujuſmodi <lb/>motus quas _Mathematici_ præſtruunt hypotheſes, ſunt hæ: </s>
  <s xml:id="echoid-s8601" xml:space="preserve">Quôd <lb/>punctum à præfixo termino in linea recta quouſque libuerit adſignare <lb/>directè progredi queat, quali motu perſpicuum eſt lineam rectam
<pb o="14" file="0192" n="207" rhead=""/>
deſcribi: </s>
  <s xml:id="echoid-s8602" xml:space="preserve">Quòd linea recta per alterius cujuſvis lineæ longitudinem ità <lb/>procedere poſſit, ut ſitum intereà parallelum perpetuò ſervet (hoc eſt <lb/>ut ipſa juxta poſitionem, quam in quolibet remporis momento ſor-<lb/>titur, parallela ſit ſibi ſecundum poſitionem ſuam in alio quovis tem-<lb/>poris momento:) </s>
  <s xml:id="echoid-s8603" xml:space="preserve">Item, quod linea quævis (definitè vel indefinitè <lb/>protenſa, quod in omnibus intelligendum) motu directo, itidem ſibi <lb/>parallelo, progredi poſſit (directo inquam, hoc eſt ut ejus ſingula <lb/>pu<unsure/>ncta lineas rectas deſcribant) qui ſanè duo motus ſibimet æquiva-<lb/>lent, eundémque procreant effectum eorúmque alterutro productæ <lb/>concipiantur illæ, quæ præ cæteris æquabiles, ac uni@ormes haberi <lb/>merentur ſuperficies; </s>
  <s xml:id="echoid-s8604" xml:space="preserve">quales ſunt in plano _Superficie par allelogramma<unsure/>_ <lb/>(ſeu penitus rectilineæ, ſive mixtæ) in Solido (ut ita dicam, vel non <lb/>in uno plano delineatæ) _Superficies priſmaticæ, Cylindricæque,_ tum <lb/>
<anchor type="note" xlink:label="note-0192-01a" xlink:href="note-0192-01"/>
quæ ſtricto, tum quæ latiori ſignificatu dicuntur. </s>
  <s xml:id="echoid-s8605" xml:space="preserve">Sit in exemplum <lb/>primò recta linea BC, cui inſiſtens recta AB per ipſam BC feratur, <lb/>ſibi continuo parallela, donec puncto B ad C promoto recta AB ipſi <lb/>DC ad AB parallelæ congruat. </s>
  <s xml:id="echoid-s8606" xml:space="preserve">Manifeſtum eſt hujuſmodi motu <lb/>procreari _figuram planam parallelogrammam_ ABCD. </s>
  <s xml:id="echoid-s8607" xml:space="preserve">Patet etiam <lb/>quodlibet aſſumptum in AB punctum, ut E, rectam lineam deſcri-<lb/>bere, cujus partes EE rectis AB interceptæ, rectæ BC partibus <lb/>BB, per eaſdem reſpectivè rectas AB interceptis (hoc eſt eodem <lb/>tempore à puncto B decurſis) æquantur. </s>
  <s xml:id="echoid-s8608" xml:space="preserve">Neque minùs patet, ſi <lb/>vice versâ recta BC per ipſam BA feratur, eandem ſuperficiem de-<lb/>lineari; </s>
  <s xml:id="echoid-s8609" xml:space="preserve">omniáque rectæ BC puncta (ceu F) rectas lineas effingere; <lb/></s>
  <s xml:id="echoid-s8610" xml:space="preserve">
<anchor type="note" xlink:label="note-0192-02a" xlink:href="note-0192-02"/>
nec non harum partes FF parallelis BC interceptas reſpectivis lineæ <lb/>AB partibus BB adæquari. </s>
  <s xml:id="echoid-s8611" xml:space="preserve">(Notetur autem abhinc brevitatis ergò <lb/>tam in his, quàm in ſimilibus caſibus harum linearum illam, quæ <lb/>motu ſuo magnitudinem deſcribit à me _Genetricem_ dici; </s>
  <s xml:id="echoid-s8612" xml:space="preserve">alteram <lb/>autem, juxta quam, vel cui inſiſtens, prior defertur, _Directricem_ ap-<lb/>pellari; </s>
  <s xml:id="echoid-s8613" xml:space="preserve">quia motæ lineæ proceſlus ab ea dirigitur, vel ad eam accom-<lb/>modatur.) </s>
  <s xml:id="echoid-s8614" xml:space="preserve">Sit rurſus linea quæpiam curva (velut arcus circularis) <lb/>
<anchor type="note" xlink:label="note-0192-03a" xlink:href="note-0192-03"/>
BC, cui in eodem plano inſiſtat linea recta AB; </s>
  <s xml:id="echoid-s8615" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8616" xml:space="preserve">per curvam BC <lb/>continuò deferatur recta AB, ſibimet æquidiſtans, donec punctum B <lb/>ad C pertigerit, &amp; </s>
  <s xml:id="echoid-s8617" xml:space="preserve">recta AB demum rectæ DC ad ipſam AB primò <lb/>poſitam parallelæ congruerit; </s>
  <s xml:id="echoid-s8618" xml:space="preserve">deſcribetur hoc motu figura quoque <lb/>plano (latiore ſignificatu) parallelogramma; </s>
  <s xml:id="echoid-s8619" xml:space="preserve">quia ſcilicet adverſa <lb/>hujus ſiguræ latera ſibi parallela ſunt, recta AB rectæ DC, &amp; </s>
  <s xml:id="echoid-s8620" xml:space="preserve">curva <lb/>AD curvæ BC. </s>
  <s xml:id="echoid-s8621" xml:space="preserve">Nam &amp; </s>
  <s xml:id="echoid-s8622" xml:space="preserve">hîc ſingula quæque _Genetricis_ rectæ puncta<unsure/> <lb/>(velut E) lineas deſcribent _directrici_ BC ſimiles &amp; </s>
  <s xml:id="echoid-s8623" xml:space="preserve">æquales; </s>
  <s xml:id="echoid-s8624" xml:space="preserve">cùm <lb/>integras, tum iiſdem parallelis AB interceptas partes; </s>
  <s xml:id="echoid-s8625" xml:space="preserve">ſi enim duo
<pb o="15" file="0193" n="208" rhead=""/>
puncta quævis EE rectâ lineâ connectantur, iíſque reſpondentia <lb/>puncta BB rectâ quoque jungantur; </s>
  <s xml:id="echoid-s8626" xml:space="preserve">quoniam rectæ EB ſibimet <lb/>æquantur (etenim nil aliud ſunt, quam eadem ipſa linea diverſum <lb/>
<anchor type="note" xlink:label="note-0193-01a" xlink:href="note-0193-01"/>
ſitum obtinens) ac parallelæ ſecundum _hypotbeſin_, erunt rectæ EE, <lb/>BB æquales ac parallelæ. </s>
  <s xml:id="echoid-s8627" xml:space="preserve">Unde patet curvas EE, BB adæquari ſi-<lb/>bimet, &amp; </s>
  <s xml:id="echoid-s8628" xml:space="preserve">aſſimilari. </s>
  <s xml:id="echoid-s8629" xml:space="preserve">Adæquari quia ſubtenſæ omnes EE ſubtenſis BB <lb/>ſingillatim æquantur; </s>
  <s xml:id="echoid-s8630" xml:space="preserve">aſſimilari, quia rectæ AB cum ſubtenſis adja-<lb/>centibus reſpectivis EE, &amp; </s>
  <s xml:id="echoid-s8631" xml:space="preserve">BB pares angulos conſtituunt, adeóque <lb/>rectæ ipſæ EE pares iis, quos rectæ BB; </s>
  <s xml:id="echoid-s8632" xml:space="preserve">ipſæ illæ cum ſeipſis, &amp; </s>
  <s xml:id="echoid-s8633" xml:space="preserve"><lb/>hæ cum ſeipſis (nam in hujuſmodi proportionalitate partium, &amp; </s>
  <s xml:id="echoid-s8634" xml:space="preserve">an-<lb/>gulorum æqualitate, ſicut alibi fortaſſe luculentiùs &amp; </s>
  <s xml:id="echoid-s8635" xml:space="preserve">fuſiùs diſſere-<lb/>mus, omnis conſiſtit linearum, &amp; </s>
  <s xml:id="echoid-s8636" xml:space="preserve">quarumcunque magnitudinum ſimi-<lb/>litudo.) </s>
  <s xml:id="echoid-s8637" xml:space="preserve">Quod ſi vice commutatâ linea curva BC fiat linea _Genetriæ<unsure/>,_ <lb/>&amp; </s>
  <s xml:id="echoid-s8638" xml:space="preserve">recta BA _directrix_, hoc eſt ſi BC per BA ſibi parallela feratur, <lb/>
<anchor type="note" xlink:label="note-0193-02a" xlink:href="note-0193-02"/>
producetur eadem ipſiſſima parallelogramma Superficies; </s>
  <s xml:id="echoid-s8639" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8640" xml:space="preserve">ſingula <lb/>rectæ BC puncta, veluti F, rectas lineas ad BA parallelas deſcri-<lb/>bent; </s>
  <s xml:id="echoid-s8641" xml:space="preserve">neque non interceptæ FF reſpectivis BB pares erunt; </s>
  <s xml:id="echoid-s8642" xml:space="preserve">quod <lb/>&amp; </s>
  <s xml:id="echoid-s8643" xml:space="preserve">pari modo ex ſuppoſito perpetuo curvæ BC paralleliſmo facilè con-<lb/>ſectatur. </s>
  <s xml:id="echoid-s8644" xml:space="preserve">Sit denique curva quævis (vel è rectis angulos efficientibus <lb/>compoſita, quæ curvæ quoque nomen meritò ferat; </s>
  <s xml:id="echoid-s8645" xml:space="preserve">_Archimedes_ <lb/>ſaltem è rectis compoſitas lineas, utì figurarum circulis inſcriptarum <lb/>aut adſcriptarum perimetros, {και} {πα}λῶν {γρ}αμμ@ν nomine complecti-<lb/>tur; </s>
  <s xml:id="echoid-s8646" xml:space="preserve">ut &amp; </s>
  <s xml:id="echoid-s8647" xml:space="preserve">viciſſim curvæ quævis lineæ cenſeri poſſunt è rectis, innu-<lb/>meris quidem illis indefinitè parvis, adjacentibus, &amp; </s>
  <s xml:id="echoid-s8648" xml:space="preserve">deinceps ſe-<lb/>cum angulos efficientibus, conſlatæ) ſit, inquam, talis aliqua curva <lb/>BC, in plano quovis conſtituta, tum in alio plano, vel ſuper lineæ <lb/>BC planum ut libet elevata, recta AB ſibi continuò feratur parallela, <lb/>modo quo ſemel ac iterum oſtendimus; </s>
  <s xml:id="echoid-s8649" xml:space="preserve">deſcribetur hujuſmodi motu <lb/>_Superficies cylindrica_ (vel certè _priſmatica, ſi linea directrix è rectis_ <lb/>ponatur compoſita) &amp; </s>
  <s xml:id="echoid-s8650" xml:space="preserve">_cylindrica_ quidem ſtrictè dicta, ſi _directrix<unsure/>_ <lb/>_fuerit linea circularis, aut elliptica_; </s>
  <s xml:id="echoid-s8651" xml:space="preserve">latiore verò ſenſu talis, ſi curva <lb/>fuerit alterius generis ut _parabolica_ puta, vel _hyperbolica_, vel alia <lb/>quæpiam. </s>
  <s xml:id="echoid-s8652" xml:space="preserve">In hoc autem motu lineæ quoque genetricis ſingula puncta <lb/>ſimiles &amp; </s>
  <s xml:id="echoid-s8653" xml:space="preserve">æquales deſcribunt curvæ directrici lineas; </s>
  <s xml:id="echoid-s8654" xml:space="preserve">æquales <lb/>(ut in mox præcedente diſcurſu) quoniam EB pares ac paral-<lb/>lelæ ſunt; </s>
  <s xml:id="echoid-s8655" xml:space="preserve">adeóque EE, BB quoque pares, ac parallelæ <lb/>ſimiles; </s>
  <s xml:id="echoid-s8656" xml:space="preserve">*quoniam etiam anguli EEE, angulis BBB æquantur. <lb/></s>
  <s xml:id="echoid-s8657" xml:space="preserve">
<anchor type="note" xlink:label="note-0193-03a" xlink:href="note-0193-03"/>
Quinetiam reciprocè deſcribatur eadem Superficies ponendo curvam <lb/>BC perrectam AB parallelωs deportari. </s>
  <s xml:id="echoid-s8658" xml:space="preserve">Quomodò ſingula quoque <lb/>curvæ BC puncta rectas parallelas &amp; </s>
  <s xml:id="echoid-s8659" xml:space="preserve">pares interceptis reſpectivis
<pb o="16" file="0194" n="209" rhead=""/>
rectæ AB partibus delineabunt, pariter ut antehac in ſiguræ planæ <lb/>exemplo commonſtratum eſt; </s>
  <s xml:id="echoid-s8660" xml:space="preserve">unde ſi ſuperſicies hoc modo procreatæ <lb/>à plano quolibet ad rectam ſeu genetricem, ſeu directricem (quam <lb/>ubique ſitam Superficiei productæ latus appellare licet) parallelo <lb/>ſecetur, ſectio communis duabus rectis parallelis conſtabit æqualibus <lb/>inter ſe. </s>
  <s xml:id="echoid-s8661" xml:space="preserve">De Superficiebus autem ità progenitis obſervatu dignum eſt <lb/>(nec enim planè nudas magnitudinum generationes indigitare, ſed &amp; </s>
  <s xml:id="echoid-s8662" xml:space="preserve"><lb/>generales nonnullas ipſarum affectiones è diverſis reſultantes generandi <lb/>modis inſinuare propoſitum eſt nobis) quòd ſi linea directrix recta ſit <lb/>(ut in figura per literam Z diſcriminata) Superficiei productæ partes <lb/>parallelis lineis genetricibus interjectæ reſpectivis directricis lineæ <lb/>partibus ſemper proportionales ſunt (ſuperficies nempe BCCB re-<lb/>ſpectivis rectis BB:) </s>
  <s xml:id="echoid-s8663" xml:space="preserve">At ſi linea curva pro directrice habeatur (ut in <lb/>figura Y) non ſemper eveniet, ut interceptæ genetricibus rectis Super-<lb/>ficies interceptis curvæ directricis partibus proportionentur; </s>
  <s xml:id="echoid-s8664" xml:space="preserve">at ſaltem <lb/>accidet hoc, cùm recta genetrix AB æqualiter ad curvam BC ubique, <lb/>vel ſecundum omnia ejus puncta inclinatur; </s>
  <s xml:id="echoid-s8665" xml:space="preserve">quomodo fit in cylindri <lb/>cujuſcunque, laxè vel ſtrictè dicti, recti ſuperficie; </s>
  <s xml:id="echoid-s8666" xml:space="preserve">quia tum recta <lb/>genetrix omnibus curvæ punctis (hoc eſt omnibus eam ad dicta puncta <lb/>
<anchor type="note" xlink:label="note-0194-01a" xlink:href="note-0194-01"/>
tangentibus, eive ſubtenſis rectis eſt perpendicularis.) </s>
  <s xml:id="echoid-s8667" xml:space="preserve">Verum ſi, in <lb/>exemplum, curva BC ponatur arcus circularis, qui dividatur æqualiter <lb/>ad puncta B, non erunt neceſſariò ſuperficies ABBA peripheriis <lb/>æqualibus BB inſiſtentes inter ſe pares, quia (præterquam in caſu <lb/>prædicto cylindri recti) rectæ AB ubique ad puncta B inæqualiter <lb/>inclinantur (unam quamvis inclinationem cum alia conferendo) an-<lb/>gulos nempe cum tangentibus ad B aliis ac aliis, &amp; </s>
  <s xml:id="echoid-s8668" xml:space="preserve">cum ſubtenſis BB <lb/>mæquales efficiunt. </s>
  <s xml:id="echoid-s8669" xml:space="preserve">E qua re pendet _inſuperabilis illa difficultas,_ <lb/>quacum conflictantur, qui _cylindricas obliquas ſuperſicies conantur_ <lb/>_dimetiri, ſen cum Cylinàricis Superficiebus rectis, aliìſve quadantenus_ <lb/>_cognitis Superficiebus quoad proportionem comparare._ </s>
  <s xml:id="echoid-s8670" xml:space="preserve">Supponunt denique <lb/>conſimili pacto ſuperſiciem quamvis planam directo motu ſibi parallelo <lb/>progredi, ſcilicet ut prædicto modo, ſingula ipſius puncta lineas <lb/>rectas deſcribant, inter ſe pares, ac parallelas; </s>
  <s xml:id="echoid-s8671" xml:space="preserve">vel ut ejus ſingulæ <lb/>rectæ (id quod indè conſectatur) planas Superficies parallelogrammas <lb/>effingant; </s>
  <s xml:id="echoid-s8672" xml:space="preserve">cujuſmodi motu deſcribuntur priſmatica quæque cylindricá-<lb/>que corpora; </s>
  <s xml:id="echoid-s8673" xml:space="preserve">illa nimirum ipſa, de quorum Superficiebus mox egimus, <lb/>quibúſque ſimili jure poſſunt adaptari, quæ Superficiebus iſtis oſtendi-<lb/>mus convenire. </s>
  <s xml:id="echoid-s8674" xml:space="preserve">Veluti quod parallelis planis interjectæ Superſicies <lb/>ipſorum, &amp; </s>
  <s xml:id="echoid-s8675" xml:space="preserve">ipſa corpora lateribus ſuis (ſeu directricis rectæ partibus <lb/>reſpectivis) proportionantur. </s>
  <s xml:id="echoid-s8676" xml:space="preserve">Quòd &amp; </s>
  <s xml:id="echoid-s8677" xml:space="preserve">ſi definita hujuſmodi corpora
<pb o="17" file="0195" n="210" rhead=""/>
planis laterum alicui parallelis ſecentur, communes ſectiones erunt <lb/>_Parallelogramma_ (quale eſt EEBB.) </s>
  <s xml:id="echoid-s8678" xml:space="preserve">Quin, ut paucis complectar <lb/>multa, quæ _de Superficiebus aut Solidis Priſmaticis ac Cylindricis_ <lb/>_ſtrictè dictis generatim enunciantur aut probantur uſpiam,_ quòd ea <lb/>pleraque juſtam analogiam obſervando, univerſis congruunt hoc modo <lb/>progenitis quantis. </s>
  <s xml:id="echoid-s8679" xml:space="preserve">Neque jam de progreſſivo motu quidpiam ſuccur-<lb/>rit adjiciendum; </s>
  <s xml:id="echoid-s8680" xml:space="preserve">quædam enim {δυ}<unsure/>σ{δί}ήγητα conſultò videntur reticen-<lb/>da. </s>
  <s xml:id="echoid-s8681" xml:space="preserve">Porrò ſimplicis motûs alterum genus, quod adhibet _Matheſis,_ <lb/>eſt _circumlatio, ſeu motus converſivus_; </s>
  <s xml:id="echoid-s8682" xml:space="preserve">qui tum ſcilicet efficitur, cùm <lb/>dimotæ magnitudinis quiddam (ut punctum aliquod puta lineæ, vel <lb/>Superficiei linea) fixum &amp; </s>
  <s xml:id="echoid-s8683" xml:space="preserve">immotum conſiſtit, dum ei velut innodata <lb/>ac adſtricta tota reliqua magnitudo, juxta quamvis aſſignatam di-<lb/>rectionem, circumagitur. </s>
  <s xml:id="echoid-s8684" xml:space="preserve">Cujuſmodi motûs generaliſſima proprie-<lb/>tas eſt, ut quæque mobilis puncta dum in uno aliquo plano tranſversè <lb/>moventur, circulares ſingula peripherias deſcribant; </s>
  <s xml:id="echoid-s8685" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8686" xml:space="preserve">quidem <lb/>omnia, quæ in eodem uno, per fixum punctum tranſeunte plano <lb/>moventur parallelas, ſeu concentricas, &amp; </s>
  <s xml:id="echoid-s8687" xml:space="preserve">ſimiles inter ſe; </s>
  <s xml:id="echoid-s8688" xml:space="preserve">quæ verò <lb/>in diverſis planis ſimiles, aut diſſimiles, prout hypotheſium exigit <lb/>arbitraria diverſitas. </s>
  <s xml:id="echoid-s8689" xml:space="preserve">Præ cæteris autem propria, maximéque na-<lb/>turalis eſt circumlatio, cùm ſingula mobilis puncta circulares unius <lb/>ejuſdem circuli peripherias deſcribunt, hoc eſt cùm in uno cuncta <lb/>plano circumferuntur; </s>
  <s xml:id="echoid-s8690" xml:space="preserve">qualem certè tum ipſa natura ſponte concipit <lb/>atque proſequitur, cùm nè rectos ſuos quos præſertim aſſectat motus <lb/>exequatur ab immobili retinaculo prohibere; </s>
  <s xml:id="echoid-s8691" xml:space="preserve">velut in pendulorum, <lb/>&amp; </s>
  <s xml:id="echoid-s8692" xml:space="preserve">libris appenſorum motibus videre eſt; </s>
  <s xml:id="echoid-s8693" xml:space="preserve">imò cùm objectâ quâvis <lb/>reſiſtentiâ non ſatìs facilè recto tramiti valet in@ærere; </s>
  <s xml:id="echoid-s8694" xml:space="preserve">ſicut in _rota-_ <lb/>_rum, &amp; </s>
  <s xml:id="echoid-s8695" xml:space="preserve">vorticum, &amp; </s>
  <s xml:id="echoid-s8696" xml:space="preserve">turbinum, &amp; </s>
  <s xml:id="echoid-s8697" xml:space="preserve">in ipſorum fortaſſe ſyderum_, <lb/>_motibuus adparet_. </s>
  <s xml:id="echoid-s8698" xml:space="preserve">Verùm hujuſmodi motuum generalem indolem <lb/>haud ità promptum eſt verbis explicare. </s>
  <s xml:id="echoid-s8699" xml:space="preserve">Præſtat ipſas quas accipiunt <lb/>præcipuas hypotheſes percenſere. </s>
  <s xml:id="echoid-s8700" xml:space="preserve">Aſſumunt primo rectam lineam in <lb/>plano circa punctum quodvis in ipſa fixum poſſe circumferri; </s>
  <s xml:id="echoid-s8701" xml:space="preserve">cujuſ-<lb/>modi motu patet omnia lineæ motæ puncta circulares peripherias de-<lb/>ſcribere; </s>
  <s xml:id="echoid-s8702" xml:space="preserve">ſingulas ab uno quovis deſcriptas ſingulis ab altero quolibet <lb/>ſimul eodem tempore deſcriptis parallelas, &amp; </s>
  <s xml:id="echoid-s8703" xml:space="preserve">ſimiles. </s>
  <s xml:id="echoid-s8704" xml:space="preserve">Ut ſi linea <lb/>recta AB manente fixo puncto C circumferatur, ſingula puncta A, <lb/>
<anchor type="note" xlink:label="note-0195-01a" xlink:href="note-0195-01"/>
E, B peripherias circulares AA, EE, BB ſibi parallelas, &amp; </s>
  <s xml:id="echoid-s8705" xml:space="preserve">ſimiles <lb/>omnes (iiſdem nimirum, aut æqualibus angulis ſubtenſas, quorum <lb/>commune centrum, aut vertex C) deſcribent. </s>
  <s xml:id="echoid-s8706" xml:space="preserve">Hoc autem modo <lb/>conſtat procreari circulos, &amp; </s>
  <s xml:id="echoid-s8707" xml:space="preserve">ſectorum circulares areas (quales ACA, <lb/>BCB,) ſed &amp; </s>
  <s xml:id="echoid-s8708" xml:space="preserve">annulos planos; </s>
  <s xml:id="echoid-s8709" xml:space="preserve">qualis eſt is qui reſtat, ſi è circulo
<pb o="18" file="0196" n="211" rhead=""/>
majore AABB detrahatur minor circulus concentricus EEEE. </s>
  <s xml:id="echoid-s8710" xml:space="preserve">E <lb/>qua geneſi colligitur circulorum, &amp; </s>
  <s xml:id="echoid-s8711" xml:space="preserve">ſectorum circularium areas, è <lb/>circularibus peripheriis, integris aut partialibus concentricis ac ſimili-<lb/>bns, conſtare tot numero qu<unsure/>ot radius puncta habet; </s>
  <s xml:id="echoid-s8712" xml:space="preserve">quarum proinde <lb/>calculum ineundo circularis areæ talis qualis dimenſio quam facillimè <lb/>reperitur; </s>
  <s xml:id="echoid-s8713" xml:space="preserve">id quod non eſt hujus temporis ulteriùs exponere. </s>
  <s xml:id="echoid-s8714" xml:space="preserve">Quin-<lb/>etiam ſupponunt lineam quamvis rectam, indeſinitè protenſam, uno <lb/>manente fixo ipſius puncto circa deſignatam quamvis in alio plano <lb/>conſtitutam lineam, curvam aut è rectis compoſitam, revolvi, ſic ut <lb/>ei nempe lineæ ſemper inſiſtat, vel eam quaſi lambat, aut perſtringat. <lb/></s>
  <s xml:id="echoid-s8715" xml:space="preserve">
<anchor type="note" xlink:label="note-0196-01a" xlink:href="note-0196-01"/>
Sit, exempli causâ, linea recta AB indefinitè protenſa, &amp; </s>
  <s xml:id="echoid-s8716" xml:space="preserve">in ea <lb/>fixum punctum V; </s>
  <s xml:id="echoid-s8717" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8718" xml:space="preserve">per V ſemper feratur linea AB juxta lineam <lb/>quamlibet BC in alio plano collocatam; </s>
  <s xml:id="echoid-s8719" xml:space="preserve">ità quidem ut aliquod lineæ <lb/>mobilis punctum continuò lineæ BC inhæreat; </s>
  <s xml:id="echoid-s8720" xml:space="preserve">ex hujuſmodi motu <lb/>producetur curva Superficies (è planis ſaltem compoſita, quam &amp; </s>
  <s xml:id="echoid-s8721" xml:space="preserve"><lb/>generali ratione, poſt _Archimedem_, curvam appellare nil vetat) <lb/>quæ quidem ſi linea directrix tota componatur è definitè magnis rectis <lb/>lineis, fiet _Superficies py@@m dalis_, è triangulis ad verticem V concur-<lb/>rentibus aggregata; </s>
  <s xml:id="echoid-s8722" xml:space="preserve">ſin circularis fuerit, aut conicarum ſectionum <lb/>aliqua, Superficies evadet ſtrictè _conica_; </s>
  <s xml:id="echoid-s8723" xml:space="preserve">ſin alterius generis aliqua, <lb/>conica ſaltem extenſo latiùs ſignificatu dicatur; </s>
  <s xml:id="echoid-s8724" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8725" xml:space="preserve">à quibuſdam di-<lb/>citur. </s>
  <s xml:id="echoid-s8726" xml:space="preserve">Cujus quidem Superficiei proprietas eſt, ex ipſa generatione <lb/>maniſeſta, quòd ſi per fixum punctum V plano ſecetur, communis <lb/>plani cum ipſa ſectio erit angulus rectilineus. </s>
  <s xml:id="echoid-s8727" xml:space="preserve">Nam ſi planum ipſam <lb/>ſecans per V lineæ directrici occurrat in punctis duobus, ut in D, E <lb/>(occurret autem in duobus, aliàs Superficiem ipſam non ſecaret) ductæ <lb/>rectæ VD, VE erunt tam in plano ſecante, quàm in curva Super-<lb/>ficie; </s>
  <s xml:id="echoid-s8728" xml:space="preserve">in plano, ex plani natura; </s>
  <s xml:id="echoid-s8729" xml:space="preserve">in Superficie, quia genetrix eadem <lb/>recta per harum terminos tranſit, ipsíſque proinde coincidit. </s>
  <s xml:id="echoid-s8730" xml:space="preserve">In hu-<lb/>juſmodi verò motu poſito quòd lineæ rectæ à puncto fixo V (ſeu ver-<lb/>tice) ad directricem lineam BC ductæ ſunt inæquales inter ſe, ſatìs <lb/>liquet lineam BC non à lineà B delineari, vel perambulari, quia <lb/>lineæ inæquales (ut VB, VE, VC) ſibi nequeunt congruere; </s>
  <s xml:id="echoid-s8731" xml:space="preserve">ade-<lb/>
<anchor type="note" xlink:label="note-0196-02a" xlink:href="note-0196-02"/>
óque punctum B progrediens ſupra, vel infra puncta B, E, C cadet; <lb/></s>
  <s xml:id="echoid-s8732" xml:space="preserve">ut nec eâdem inæqualitate ſuppoſitâ punctum quodvis aliud in VB puta <lb/>G) motu ſuo lineam deſcribet lineæ directrici BC ſimilem (quare <lb/>linea VB ſupponitur indefinitè protenſa) at verò ſi lineæ omnes, quæ <lb/>ab V ad BC duci poſſunt (quas Superficiei propoſitæ latera nuncu-<lb/>pemus licet) proportionaliter ſecentur (id quod fiet à plano per hanc <lb/>Superſiciem trajecto ad planum, in quo ſita eſt BC, parallelo) divi-
<pb o="19" file="0197" n="212" rhead=""/>
ſionum puncta lineam conſtituent, ſaltem ad lineam conſiſtent, ipſi <lb/>BC ſimilem. </s>
  <s xml:id="echoid-s8733" xml:space="preserve">Ductis enim quotlibet lateribus VB, VD, VE, VC, <lb/>&amp; </s>
  <s xml:id="echoid-s8734" xml:space="preserve">ducto plano GKLH ad planum BDEC parallelo, ſint com-<lb/>
<anchor type="note" xlink:label="note-0197-01a" xlink:href="note-0197-01"/>
munes plani VBD cum planis BC, GH ſectiones rectæ BD, CH; <lb/></s>
  <s xml:id="echoid-s8735" xml:space="preserve">hæ parallelæ erunt. </s>
  <s xml:id="echoid-s8736" xml:space="preserve">Item communes plani VDE cum iiſdem planis <lb/>BC, GH Sectiones DE, KL parallelæ erunt. </s>
  <s xml:id="echoid-s8737" xml:space="preserve">Ergò anguli BDE, <lb/>GKL ſunt æquales. </s>
  <s xml:id="echoid-s8738" xml:space="preserve">Item ſe habet recta BD ad GK, ut DE ad <lb/>
<anchor type="note" xlink:label="note-0197-02a" xlink:href="note-0197-02"/>
KL, quia utraque hæc proportio æqualis eſt illi, quam habet VD <lb/>ad VK (ſimilia quippe ſunt triangula VDB, VKG, &amp; </s>
  <s xml:id="echoid-s8739" xml:space="preserve">triangula <lb/>VDE, VKL) permutandóque BD. </s>
  <s xml:id="echoid-s8740" xml:space="preserve">DE:</s>
  <s xml:id="echoid-s8741" xml:space="preserve">: GK. </s>
  <s xml:id="echoid-s8742" xml:space="preserve">KL. </s>
  <s xml:id="echoid-s8743" xml:space="preserve">ergò omnes <lb/>ſubtenſæ in GH proportionales ſunt ſubtenſis omnibus in BC, eas <lb/>nimirum in utraque linea ordinatim &amp; </s>
  <s xml:id="echoid-s8744" xml:space="preserve">deinceps accipiendo; </s>
  <s xml:id="echoid-s8745" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8746" xml:space="preserve">quæ <lb/>ſibimet adjacent in una pariter inflectuntur cum iis, quæ ſibi adjacent <lb/>in altera. </s>
  <s xml:id="echoid-s8747" xml:space="preserve">Ergò ſecundum ſuperiùs inſinuata lineas BC, GH ſimiles <lb/>eſſe conſtat.</s>
  <s xml:id="echoid-s8748" xml:space="preserve">‖ Hinc etiam patet lineas curvas ſimiles BC, GH ean-<lb/>dem ad ſe proportionem habere, quam Superficierum, in eadem <lb/>qualibet recta ſita, latera VB, VG. </s>
  <s xml:id="echoid-s8749" xml:space="preserve">Quum enim ſubtenſarum <lb/>iiſdem angulis incluſarum (ut BD, GK, vel DE, KL) ſingulæ <lb/>rationes æquales ſint rationi laterum VB, VG; </s>
  <s xml:id="echoid-s8750" xml:space="preserve">etiam omnes ante-<lb/>
<anchor type="note" xlink:label="note-0197-03a" xlink:href="note-0197-03"/>
cedentes conjunctæ (hoc eſt tota BC) ad omnes conſequentes con-<lb/>junctas (hoc eſt totam GH) ſe habebunt ut VB ad VG. </s>
  <s xml:id="echoid-s8751" xml:space="preserve">Hinc etiam <lb/>tali motu productarum ſuperficierum emergit hæc proprietas; </s>
  <s xml:id="echoid-s8752" xml:space="preserve">quòd <lb/>interceptæ ſcilicet à parallelis ad BC planis, à vertice deſumptæ, <lb/>quibuſcunque lateribus iiſdem incluſæ partes ipſarum ſint inter ſe ſi-<lb/>miles; </s>
  <s xml:id="echoid-s8753" xml:space="preserve">ut puta Superficies BVC, GVH; </s>
  <s xml:id="echoid-s8754" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8755" xml:space="preserve">BVD, GVK. <lb/></s>
  <s xml:id="echoid-s8756" xml:space="preserve">(Quod ex generali ſimilitudinis doctrina poſthac explicanda luculen-<lb/>tiùs apparere poterit; </s>
  <s xml:id="echoid-s8757" xml:space="preserve">interim ex ſimilitudine linearum curvarum, &amp; </s>
  <s xml:id="echoid-s8758" xml:space="preserve"><lb/>earum cum Superficiei lateribus analogia, penitúſque conſimili Superſi-<lb/>cierum generatione ſatìs eluceſcit; </s>
  <s xml:id="echoid-s8759" xml:space="preserve">ſaltem ex triangulorum VBD, <lb/>VGK; </s>
  <s xml:id="echoid-s8760" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8761" xml:space="preserve">VDE, VKL, &amp; </s>
  <s xml:id="echoid-s8762" xml:space="preserve">talium omnium ſimilitudine ſatìs con-<lb/>ſtat; </s>
  <s xml:id="echoid-s8763" xml:space="preserve">ſiquidem ex talibus infinitis triangulis utraque Superficies com-<lb/>poſita cenſeatur.) </s>
  <s xml:id="echoid-s8764" xml:space="preserve">Unde ſimilium Superficierum proprietates iis con-<lb/>venient. </s>
  <s xml:id="echoid-s8765" xml:space="preserve">Verùm quòd interceptas attinet à diverſis lateribus Super-<lb/>ficies, eas inter ſe comparando, notandum eſt quòd baſibus ſuis, ſeu <lb/>directricis lineæ reſpectivis partibus non ſemper proportionales ſunt; </s>
  <s xml:id="echoid-s8766" xml:space="preserve"><lb/>at ſaltem hoc tum evenit, cùm omnia dictæ Superficiei latera ſunt <lb/>æqualia inter ſe, adeóque cùm linea directrix eſt peripheria circuli; </s>
  <s xml:id="echoid-s8767" xml:space="preserve"><lb/>quo caſu producta Superficies erit conica Superficies ſtrictè dicta, <lb/>rectúmque quidem ad conum pertinens. </s>
  <s xml:id="echoid-s8768" xml:space="preserve">Quod ſi directrix BC ſup-<lb/>ponatur e. </s>
  <s xml:id="echoid-s8769" xml:space="preserve">g. </s>
  <s xml:id="echoid-s8770" xml:space="preserve">peripheria circularis, lateráque ſibimer inæqualia, ſi
<pb o="20" file="0198" n="213" rhead=""/>
dividatur BC in partes æquales, &amp; </s>
  <s xml:id="echoid-s8771" xml:space="preserve">connectantur latera VD, VE <lb/>non erunt Superficies BVD, DVE, EVC æquales inter ſe, ſed <lb/>inſerutabili plerumque ratione; </s>
  <s xml:id="echoid-s8772" xml:space="preserve">juxta varias angulorum incluſorum, <lb/>&amp; </s>
  <s xml:id="echoid-s8773" xml:space="preserve">laterum inæqualium differentias, inæquales; </s>
  <s xml:id="echoid-s8774" xml:space="preserve">id quod hactenus <lb/>ill<unsure/>o<unsure/>s divexavit &amp; </s>
  <s xml:id="echoid-s8775" xml:space="preserve">torſit, _qui dimetiendæ coni ſcaleni ſuperficiei incu-_ <lb/>buerunt.</s>
  <s xml:id="echoid-s8776" xml:space="preserve">‖ Ex his conſectatur quòd poſſit hujuſmodi circumlatio <lb/>facta quadantenus concipi motu quoque tali lineæ rectæ genetricis, <lb/>ità ut ejus ſingula quæque puncta parallelωs lata ſimiles directrici lineæ <lb/>lineas deſcribant, modo tamen concipiatur linea genetrix ubique pro-<lb/>portionaliter aut contrahi, vel dilatari ſecundum omnes ſui partes. <lb/></s>
  <s xml:id="echoid-s8777" xml:space="preserve">Quomodo nempe ſi recta VB ita ſenſim diduci concipiatur, ut pun-<lb/>ctum B totam lineam BC perambulet, etiam punctum G parallelo-<lb/>ad BC motu delata, lineam GH ipſi BC ſimilem deſcribet. </s>
  <s xml:id="echoid-s8778" xml:space="preserve">Qui-<lb/>nimò ſi conſimili pacto curva BC, directo quoad lineam rectam BV <lb/>motu ſitúque ſemper ad ſeipſam parallelo concipiatur promoveri, ſic <lb/>ut ejus ſingula quæque puncta lineas rectas deſcribant, ſecum omnes <lb/>in punctum V concurrentes; </s>
  <s xml:id="echoid-s8779" xml:space="preserve">hoc eſt ità ut ipſa per totum ſuum pro-<lb/>greſſum juxta ſuas omnes partes analogicè contrahatur, ad verticem <lb/>uſque V; </s>
  <s xml:id="echoid-s8780" xml:space="preserve">producentur ex hujuſmodi motibus Superficies conicæ pror-<lb/>ſus eædem cum jam proximè tractatis. </s>
  <s xml:id="echoid-s8781" xml:space="preserve">Verùm hujuſmodi motus ima-<lb/>ginarii ſunt, &amp; </s>
  <s xml:id="echoid-s8782" xml:space="preserve">quales rerum natura reſpuit. </s>
  <s xml:id="echoid-s8783" xml:space="preserve">Explicandæ tamen <lb/>hujuſmodi Superficierum naturæ deſervire poſſunt, &amp; </s>
  <s xml:id="echoid-s8784" xml:space="preserve">ſupponi ſaltem <lb/>ut per _divinam potentiam effectibiles._</s>
  <s xml:id="echoid-s8785" xml:space="preserve">‖ Ad hæc, ſi _linea directrix_ <lb/>in motu proximè memorato ſupponatur undique clauſa, ſic ut figuram <lb/>quamvis comprehendat, Superficies curva progenita cum hac figura, <lb/>ceu baſe, corpus ſolidum includet pyramidale, vel conicum (ſtrictè <lb/>vel laxè pro dictæ figuræ natura ſumptum) cujus generalia ſympto-<lb/>mata ſatìs è dictis eluceſcunt. </s>
  <s xml:id="echoid-s8786" xml:space="preserve">Nempe quòd à parallelis ad hujuſce <lb/>ſolidi baſin planis abſcindentur ſimiles ad verticem Superficies, ſimiléſ-<lb/>que baſes intercipientur, &amp; </s>
  <s xml:id="echoid-s8787" xml:space="preserve">ſimilia corpora Solida progignentur. </s>
  <s xml:id="echoid-s8788" xml:space="preserve"><lb/>Verbo dicam, quæ de _Conis_ generatim _E@clides, Apollonius,_ aliique <lb/>tradiderunt, ea conicis ho@ modo factis, ſervatâ debitâ analogia, <lb/>convenient, &amp; </s>
  <s xml:id="echoid-s8789" xml:space="preserve">ſimili ferme modo demonſtrabuntur convenire.</s>
  <s xml:id="echoid-s8790" xml:space="preserve">‖ Ve-<lb/>rùm uſitatiſſimus apud Mathematicos corpora progignendi modus eſt <lb/>is qui peculiari nomine _Rotatio_ dicitur, &amp; </s>
  <s xml:id="echoid-s8791" xml:space="preserve">fit ſuppoſito lineam quam-<lb/>vis, aut quamlibet Superſiciem planam cirea rectam lineam fixam, <lb/>tanquam axem, revolvi. </s>
  <s xml:id="echoid-s8792" xml:space="preserve">Quomodo ex motu Semiperipheriæ circu-<lb/>laris circa diametrum producitur _Sphærica Superſicies,_ ex motu Se-<lb/>micirculi ipſius circa eundem _Sphæra_ detornatur; </s>
  <s xml:id="echoid-s8793" xml:space="preserve">ex motu lineæ rectæ <lb/>circa lineam ipſi parallelam _Superſicies Cylindrica;_ </s>
  <s xml:id="echoid-s8794" xml:space="preserve">ex motu parallelo-<lb/>grammi rectanguli circa latus unum ipſé _Cylindrus rectus;_ </s>
  <s xml:id="echoid-s8795" xml:space="preserve">ex motu cruris
<pb o="21" file="0199" n="214" rhead=""/>
unius anguli rectilinei circa alterum _Conica Superficies_; </s>
  <s xml:id="echoid-s8796" xml:space="preserve">ex rectanguli <lb/>trianguli circa crus unum anguli recti _conus_ ipſe deſormatur; </s>
  <s xml:id="echoid-s8797" xml:space="preserve">eóque <lb/>pacto _cùm integræ cum ſuis Curvis Superficiebus Solidæ magnitudines_ <lb/>_innumeræ, tumipſarum portiones, fruſta, tubi, annuli procreantur._ <lb/></s>
  <s xml:id="echoid-s8798" xml:space="preserve">Cujuſmodi motûs hæc præcipua proprietas eſt, quòd ſingula quæque <lb/>magnitudinis circumductæ puncta peripherias obeant circulares (inte-<lb/>gras quidem illas, modò perfecta ſit revolutio, ſeu mobile denuo <lb/>primum in ſitum reſtituatur, at ſimiles utcunque ſibi mutuo, quæ <lb/>ſimul deſcribuntur) quarum omnia Centra ſunt in dicto axe, radii <lb/>verò ſunt rectæ ab ipſis punctis ad axem perpendiculares. </s>
  <s xml:id="echoid-s8799" xml:space="preserve">Vel; </s>
  <s xml:id="echoid-s8800" xml:space="preserve"><lb/>quod omnes in mobili ſitæ rectæ lineæ axi perpendiculares eſſiciunt <lb/>circulos (ſi revolutio ponatur integrè peracta) aut circulares ſimiles <lb/>ſectores, illos intelligo qui ſimul eodem tempore delineantur. </s>
  <s xml:id="echoid-s8801" xml:space="preserve">Ut ſi <lb/>
<anchor type="note" xlink:label="note-0199-01a" xlink:href="note-0199-01"/>
v.</s>
  <s xml:id="echoid-s8802" xml:space="preserve">g. </s>
  <s xml:id="echoid-s8803" xml:space="preserve">linea quævis circa axem VK rotetur, eo procreabitur motu curva <lb/>quædam Superficies, circularibus quaſi peripheriis conſtans (_Ato-_ <lb/>_miſtarum_ enim phraſin facilitatis, perſpicuitatis, brevitatis, addere <lb/>licet &amp; </s>
  <s xml:id="echoid-s8804" xml:space="preserve">veriſimilitudinis causâ non illibenter uſurpo) circularibus, in-<lb/>quam, peripheriis AY, BY, CY, DY per puncta A, B, C, D reli-<lb/>quáque quæ ſunt in VD cuncta decircinatis; </s>
  <s xml:id="echoid-s8805" xml:space="preserve">quarum radii ſunt rectæ <lb/>AZ, BZ, CZ, DZ axi perpendiculares, &amp; </s>
  <s xml:id="echoid-s8806" xml:space="preserve">Centra Z in axe. <lb/></s>
  <s xml:id="echoid-s8807" xml:space="preserve">Quód ſi revolutio tantum eouſque continuatur, donec VAD ſit in <lb/>ſitu V αδ, conſtabit _effecta superficies_ ex arcubus A α, B ε<unsure/>, C γ, <lb/>D δ, ſimilibus inter ſe eodem modo ſi planum VDZ circa axem <lb/>VK revolvatur, poſito quòd integra peragatur converſio, produ-<lb/>cetur Solidum quali conſtans innumeris circulis parallelis AY, BY, <lb/>CY, DY, quorum (ut priùs) radii AZ, BZ, CZ, DZ, <lb/>centra Z; </s>
  <s xml:id="echoid-s8808" xml:space="preserve">poſitóque quod circulatio deſiſtit in ſitu δ υ K, conſtitue-<lb/>tur Solidum è Sectoribus AZ α, BZ ε<unsure/>, CZ γ, &amp; </s>
  <s xml:id="echoid-s8809" xml:space="preserve">reliquis inter ſe <lb/>ſimilibus. </s>
  <s xml:id="echoid-s8810" xml:space="preserve">Cæterúm prætermittenda non eſt animadverſio quædam <lb/>perquam utilis, &amp; </s>
  <s xml:id="echoid-s8811" xml:space="preserve">neceſſaria circa _modum Superficierum, &amp; </s>
  <s xml:id="echoid-s8812" xml:space="preserve">Soli-_ <lb/>_dorum hoc modo reſultantium dimenſiones inveſtigandi juxta metbodum_ <lb/>_indiviſibilium, omnium expeditiſſimam, &amp; </s>
  <s xml:id="echoid-s8813" xml:space="preserve">modò ritè adhibeatur haud_ <lb/>_minùs certam &amp; </s>
  <s xml:id="echoid-s8814" xml:space="preserve">infallibilem._ </s>
  <s xml:id="echoid-s8815" xml:space="preserve">Objicit huic methodo non ſemel, in <lb/>pererudito ſuo de _Solidis cylindricis ac annularibus libello, do<unsure/>ctiſſimus_ <lb/>_A. </s>
  <s xml:id="echoid-s8816" xml:space="preserve">Tacquetus_, eóque ſe putat illam deſtruere, quòd per eam in-<lb/>ventæ _conorum, &amp; </s>
  <s xml:id="echoid-s8817" xml:space="preserve">Spherarum ſuperſicies_ (quantitates horum intelligo) <lb/>veræ per _Archimedem_ repertæ ac traditæ dimenſioni non reſpondent. </s>
  <s xml:id="echoid-s8818" xml:space="preserve"><lb/>Sit exemplo _rectus conus_ DVY, cujus axis VK; </s>
  <s xml:id="echoid-s8819" xml:space="preserve">per cujus omnia <lb/>puncta tranſire concipiantur axi perpendiculares rectæ ZA, ZB, <lb/>ZC, | ZD, &amp;</s>
  <s xml:id="echoid-s8820" xml:space="preserve">c. </s>
  <s xml:id="echoid-s8821" xml:space="preserve">è quibus nempe juxta _methodum atomicam_ com-<lb/>
<anchor type="note" xlink:label="note-0199-02a" xlink:href="note-0199-02"/>
| K<unsure/>
<pb o="22" file="0200" n="215" rhead=""/>
ponitur ipſum _triangulum rectangulum_ VKD; </s>
  <s xml:id="echoid-s8822" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8823" xml:space="preserve">è circulis ad quas <lb/>ceu radios deſcriptis ipſe _conus_ conflatur. </s>
  <s xml:id="echoid-s8824" xml:space="preserve">Ergò, diſputat, ex ho-<lb/>rum circulorum peripheriis _Superficies conica_ componetur; </s>
  <s xml:id="echoid-s8825" xml:space="preserve">qnod <lb/>tamen veritati comperitur adverſari; </s>
  <s xml:id="echoid-s8826" xml:space="preserve">methodúſque proinde fallax <lb/>eſt. </s>
  <s xml:id="echoid-s8827" xml:space="preserve">Repono, malè calculum hoc pacto iniri; </s>
  <s xml:id="echoid-s8828" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8829" xml:space="preserve">in peripheriarum è <lb/>quibus _Superficies_ conſtant computatione diverſam inſtituendam eſſe <lb/>rationem ab ea, quâ computantur lineæ quibus _planæ ſuperficies_ con-<lb/>ſtant, aut plana, è quibus corpora formantur. </s>
  <s xml:id="echoid-s8830" xml:space="preserve">Nempe peripheria-<lb/>rum Superficiem curvam conſtituentium è revolutione prognatam <lb/>lineæ VD cenſeri debet è multitudine punctorum, quæ ſunt in ipſa <lb/>
<anchor type="note" xlink:label="note-0200-01a" xlink:href="note-0200-01"/>
linea genetrice VD; </s>
  <s xml:id="echoid-s8831" xml:space="preserve">quippe cùm per ea ſingula puncta tales peri-<lb/>pheriæ tranſeant, nec plures tranſire queant; </s>
  <s xml:id="echoid-s8832" xml:space="preserve">quicunque ſit axis, ſeu <lb/>longiùs diſtans, ſeu propiùs adjacens; </s>
  <s xml:id="echoid-s8833" xml:space="preserve">axis enim ſolummodò, pro <lb/>longiore vel propiore diſtantia poſitionéque varia, dictarum periphe-<lb/>riarum magnitudinem determinat. </s>
  <s xml:id="echoid-s8834" xml:space="preserve">Verùm multitudo linearum ex <lb/>quibus planum DVK ſupponitur conſtare, planorúmque quibus <lb/>Solidum DVY conſtat, è numero taxanda eſt punctorum in axe <lb/>VK; </s>
  <s xml:id="echoid-s8835" xml:space="preserve">nec enim plures intra terminos VK parallelæ, ipſi VK perpen-<lb/>diculares, rectæ, vel plura talia parallela plana duci poſſunt, quam <lb/>horum punctorum multitudini æquinumera. </s>
  <s xml:id="echoid-s8836" xml:space="preserve">Quod obſervando _diſcri-_ <lb/>_men_ (ſedulò perpendendum) omnem devitabimus errorem, &amp; </s>
  <s xml:id="echoid-s8837" xml:space="preserve">_cur-_ <lb/>_varum bujuſmodi rot@tu genitarum Superficierum facillimo, reor,_ <lb/>_omnium quos rei natura ſubminiſtr at modo perquiremus._ </s>
  <s xml:id="echoid-s8838" xml:space="preserve">Illum com-<lb/>monſtrabo. </s>
  <s xml:id="echoid-s8839" xml:space="preserve">Pro reperienda v. </s>
  <s xml:id="echoid-s8840" xml:space="preserve">g. </s>
  <s xml:id="echoid-s8841" xml:space="preserve">dimenſione _curvæ ſuperficiei_ lineæ <lb/>VD circa axem VK revolutione, concipiatur ipſa VD in directum <lb/>extendi, ità ſcilicet ut ei exæquetur recta VD; </s>
  <s xml:id="echoid-s8842" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8843" xml:space="preserve">ad ejus omnia <lb/>puncta rectæ concipiantur applicari ipſi VD perpendiculares, &amp; </s>
  <s xml:id="echoid-s8844" xml:space="preserve">pe-<lb/>ripheriis circularibus, è quibus Superficies curva conflatur, ordine <lb/>pares; </s>
  <s xml:id="echoid-s8845" xml:space="preserve">ſingulæ ſingulis, puta AX ipſi AY, &amp; </s>
  <s xml:id="echoid-s8846" xml:space="preserve">CX ipſi CY, ac <lb/>ità continuò. </s>
  <s xml:id="echoid-s8847" xml:space="preserve">Erit ex his parallelis rectis conſtitutum planum VDX <lb/>æquale _dictæ curvæ ſuperficiei;_ </s>
  <s xml:id="echoid-s8848" xml:space="preserve">hujúſque partes illius partibus re-<lb/>ſpectivis. </s>
  <s xml:id="echoid-s8849" xml:space="preserve">Sin loco _peripberiarum_ applicentur ipſarum reſpectivi radii <lb/>AZ, BZ, CZ, &amp; </s>
  <s xml:id="echoid-s8850" xml:space="preserve">reliqui; </s>
  <s xml:id="echoid-s8851" xml:space="preserve">ſpatium ex his rectis conſtitutum (quæ <lb/>ſanè proportionali cum alteris ſerie procedunt) ſe habebit ad _curvam_ <lb/>_Superficiem, ut c@rculi cujuſvis radius ad ejus circumſerentiam._ </s>
  <s xml:id="echoid-s8852" xml:space="preserve">Un-<lb/>de ſiquâ ratione deprehendi poſſit _Summa radiorum peromnia lineæ_ <lb/>_genetricis puncta tranſeuntium (hoc eſt ſi ſpatii VDZ dimenſionem_ <lb/>reperire contigerit) eo ſtatim innoteſcet _curvæ Superſiciei dimenſio._ <lb/></s>
  <s xml:id="echoid-s8853" xml:space="preserve">In exemplum, facilitatis ergò, proponatur _conica Superficies_ DVY, <lb/>è rotatu procreata rectæ VD, circa axem VK. </s>
  <s xml:id="echoid-s8854" xml:space="preserve">Ad rectam VD ap-
<pb o="23" file="0201" n="216" rhead=""/>
plicentur rectæ AZ, BZ, CZ, DZ, ad ipſam VD perpendiculares, <lb/>&amp; </s>
  <s xml:id="echoid-s8855" xml:space="preserve">æquales ſingulæ ſingulis in cono circulorum radiis per eaſdem li-<lb/>teras deſignatis; </s>
  <s xml:id="echoid-s8856" xml:space="preserve">fiet autem in hoc caſu _Spatium_ VDZ triangulum, <lb/>quia rectæ AZ, BZ, CZ æqualiter à ſe diſtantes æqualiter increſcunt, <lb/>id quod trianguli applicatis omninò proprium eſt. </s>
  <s xml:id="echoid-s8857" xml:space="preserve">Hujus autem <lb/>trianguli, ex datis altitudine VD &amp; </s>
  <s xml:id="echoid-s8858" xml:space="preserve">baſe DZ, dimenſio in promptu <lb/>eſt. </s>
  <s xml:id="echoid-s8859" xml:space="preserve">Quod ſi ſiat _ut circuli radius: </s>
  <s xml:id="echoid-s8860" xml:space="preserve">Ad circumferentiam ipſius, it à_ <lb/>_triangulum VDZ ad quartum_, erit hoc quartum æquale _Superficiei_ <lb/>_conicæ propoſitæ._ </s>
  <s xml:id="echoid-s8861" xml:space="preserve">Eodem planè modo perquam facilè _Sphæræ<unsure/>, sphæri-_ <lb/>_carúmque portionum Superficies_ (nec, datis &amp; </s>
  <s xml:id="echoid-s8862" xml:space="preserve">præcognitis iis quæ <lb/>requiruntur, alias quaſlibet hoc modo natas) inveſtigare licet. </s>
  <s xml:id="echoid-s8863" xml:space="preserve">At <lb/>mihi propoſitum eſt generalioribus tantùm inhærere. </s>
  <s xml:id="echoid-s8864" xml:space="preserve">‖ Hanc autem <lb/>magnitudinum geneſin æmulatur, &amp; </s>
  <s xml:id="echoid-s8865" xml:space="preserve">affinitate quâdam contingit iſte <lb/>modus, quum circa rectam lineam, (aut quidem circa quamvis <lb/>aliam) ſimiles innumeræ lineæ, vel figuræ parallelo juxta ſe ſitu <lb/>diſpoſitæ talit<unsure/>er conſtituuntur, ut ſingulæ centrum ſuum habeant in <lb/>dicta linea, quæ proinde tanquam axis rationem ſubit, ac talis deno-<lb/>minatur. </s>
  <s xml:id="echoid-s8866" xml:space="preserve">Quomodo, e. </s>
  <s xml:id="echoid-s8867" xml:space="preserve">c, in _cylindris obliquis_, ínque _conis Scalenis_ <lb/>circuli circa lineam quandam rectam conſiſtunt; </s>
  <s xml:id="echoid-s8868" xml:space="preserve">quæ proptereà dicitur <lb/>ipſorum _axis_, quoniam in ea circulorum parallelorum _centra_ ex-<lb/>iſtunt. </s>
  <s xml:id="echoid-s8869" xml:space="preserve">Sed cùm motus ità diſtortos natura non capiat (ſaltem juxta <lb/>modum operandi ſimplicem quem nunc ſupponimus) &amp; </s>
  <s xml:id="echoid-s8870" xml:space="preserve">quia poſſunt <lb/>hujuſmodi magnitudines ut modis aliis genitæ faciliùs concipi, de iis <lb/>abſtinebimus. </s>
  <s xml:id="echoid-s8871" xml:space="preserve">Neque non de magnitudinum per motus ſimplices <lb/>effectione ſufficiet hactenus diſſeruiſſe.</s>
  <s xml:id="echoid-s8872" xml:space="preserve">‖</s>
</p>
<div xml:id="echoid-div218" type="float" level="2" n="1">
<note position="left" xlink:label="note-0192-01" xlink:href="note-0192-01a" xml:space="preserve">Fig. 3.</note>
<note position="left" xlink:label="note-0192-02" xlink:href="note-0192-02a" xml:space="preserve">Fig. 4.</note>
<note position="left" xlink:label="note-0192-03" xlink:href="note-0192-03a" xml:space="preserve">Fig. 5.</note>
<note position="right" xlink:label="note-0193-01" xlink:href="note-0193-01a" xml:space="preserve">Fig. 5.</note>
<note position="right" xlink:label="note-0193-02" xlink:href="note-0193-02a" xml:space="preserve">Fig. 6.</note>
<note position="right" xlink:label="note-0193-03" xlink:href="note-0193-03a" xml:space="preserve">10. XI<unsure/>. El<unsure/>@m.</note>
<note position="left" xlink:label="note-0194-01" xlink:href="note-0194-01a" xml:space="preserve">Fig. 6.</note>
<note position="right" xlink:label="note-0195-01" xlink:href="note-0195-01a" xml:space="preserve">Fig. 7</note>
<note position="left" xlink:label="note-0196-01" xlink:href="note-0196-01a" xml:space="preserve">Fig. 8.</note>
<note position="left" xlink:label="note-0196-02" xlink:href="note-0196-02a" xml:space="preserve">Fig. 8.</note>
<note position="right" xlink:label="note-0197-01" xlink:href="note-0197-01a" xml:space="preserve">16. XI.Elem.</note>
<note position="right" xlink:label="note-0197-02" xlink:href="note-0197-02a" xml:space="preserve">10. XI. elen<unsure/>.</note>
<note position="right" xlink:label="note-0197-03" xlink:href="note-0197-03a" xml:space="preserve">12. V. Elem.</note>
<note position="right" xlink:label="note-0199-01" xlink:href="note-0199-01a" xml:space="preserve">Fig. 9.</note>
<note position="right" xlink:label="note-0199-02" xlink:href="note-0199-02a" xml:space="preserve">Fig. 10, 11, 12.</note>
<note position="left" xlink:label="note-0200-01" xlink:href="note-0200-01a" xml:space="preserve">Fig. 10, 11, 12.</note>
</div>
<pb o="24" file="0202" n="217"/>
</div>
<div xml:id="echoid-div220" type="section" level="1" n="30">
<head xml:id="echoid-head33" xml:space="preserve"><emph style="sc">Lect</emph>. III.</head>
<p>
  <s xml:id="echoid-s8873" xml:space="preserve">Q Uomodo per _motus ſimplices progreſſivum, &amp; </s>
  <s xml:id="echoid-s8874" xml:space="preserve">converſivum_ <lb/>_ffectæ concipiantur magnitudines, &amp; </s>
  <s xml:id="echoid-s8875" xml:space="preserve">qualia generationes iſtas_ <lb/>_conſequuntur ſymptomata_ (nonnulla ſaltem præcipua) _con-_ <lb/>_niſi ſumus exponere ad compoſitos nunc, &amp; </s>
  <s xml:id="echoid-s8876" xml:space="preserve">concurrentes,_ <lb/>_eidem propoſito ſervientes, motns accingimur;_ </s>
  <s xml:id="echoid-s8877" xml:space="preserve">quorum in effectis <lb/>diſcernendis velocitates, ſecundum quas ſimplices peraguntur motus, <lb/>omnino, vel cum primis conſiderandæ ſunt; </s>
  <s xml:id="echoid-s8878" xml:space="preserve">quarum in generatione <lb/>per motus ſimplices nulla prorſus habetur ratio. </s>
  <s xml:id="echoid-s8879" xml:space="preserve">Per eundem enim <lb/>motum ſimplicem ſeu velocior is ſit, ſeu tardior eadem magnitudo, <lb/>quamvìs non eodem temporis intervallo, producitur; </s>
  <s xml:id="echoid-s8880" xml:space="preserve">idem nempe <lb/>_circulus_ ex ejuſdem rectæ circa punctum in ea fixum, _eadem Sphæra_ <lb/>ex Semicirculi circa _diametrum_ rotatu; </s>
  <s xml:id="echoid-s8881" xml:space="preserve">quamvìs ut hæc fiant eò ma-<lb/>gìs awt minus expectandum ſit, quo ſegnior aut citatior ſupponitur ea <lb/>progenerans motus. </s>
  <s xml:id="echoid-s8882" xml:space="preserve">Verùm in generatione per motus compoſitos <lb/>iiſdem manentibus lationis modis, prout unius aut plurium variatur <lb/>velocitas, nedum ſpecie, ſed etiam quantitate diverſæ magnitudines <lb/>emergere ſolent, poſitione ſaltem perpetuò differentes. </s>
  <s xml:id="echoid-s8883" xml:space="preserve">Ut ſi recta <lb/>
<anchor type="note" xlink:label="note-0202-01a" xlink:href="note-0202-01"/>
AB per rectam AC parallelo deferatur æquabili motu; </s>
  <s xml:id="echoid-s8884" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8885" xml:space="preserve">ſimul <lb/>punctum M in AB deſcendat uniformiter; </s>
  <s xml:id="echoid-s8886" xml:space="preserve">vel ſimulrecta AC pa-<lb/>rallelo quoque uniformi motu deſcendens ipſam AB promotam inter-<lb/>ſecet in M; </s>
  <s xml:id="echoid-s8887" xml:space="preserve">ex ejuſmodi motuum compoſitione vel concurſu produ-<lb/>cetur recta linea AM. </s>
  <s xml:id="echoid-s8888" xml:space="preserve">Quòd ſi eodem, etiam quoad velocitatem <lb/>manente motu rectæ AB, immutetur in velocitate motus uniformis <lb/>puncti M, vel rectæ AC, ità quidem punctum M jam eodem tem-<lb/>pore pervenerit ad μ, vel AC ſecet ipſam AB in μ, deſcribetur <lb/>hoc motu alia recta A μ à priore AM poſitione diverſa. </s>
  <s xml:id="echoid-s8889" xml:space="preserve">Sin verò, <lb/>manente rurſus eodem motu rectæ AB, pro motu puncti M, vel <lb/>rectæ AC uniformi ſubſtituatur motus, quem vocant, æqualiter <lb/>acceleratus, ex ejuſmodi compoſitione, vel concurſu fiet linea
<pb o="25" file="0203" n="218" rhead=""/>
parabolica AMX vel etiam aliter poſita A μ Y (prout hic motus ac-<lb/>celeratus gradu ponitur alius ac alius.) </s>
  <s xml:id="echoid-s8890" xml:space="preserve">Quòd ſi quâpiam aliâ ratione <lb/>creſcere concipiatur, aut minui dicti puncti vel lineæ velocitas alia pro-<lb/>gignetur inde, pro ratione _bypotbeſis_, diverſa ſpecies magnitudinis. </s>
  <s xml:id="echoid-s8891" xml:space="preserve">In his <lb/>conſpicitur exemplis quòd eodem ſubinde recidant _compoſitio motuum_ <lb/>_et concurſus_; </s>
  <s xml:id="echoid-s8892" xml:space="preserve">quod exinde quidem contingit, quia rectæ cujuſpiam paral-<lb/>lelo motu latæſingula puncta rectas deſcribunt ſibi parallelas; </s>
  <s xml:id="echoid-s8893" xml:space="preserve">unde fit ut <lb/>perinde ſit an punctum ejus aliquod in ipſa fixum deferatur cum ea, vel <lb/>ſolutum per lineam ejus directioni parallelam; </s>
  <s xml:id="echoid-s8894" xml:space="preserve">ut nempe utrùm punctum <lb/>M in AC fixum cum ea deferatur, an liberè decurrat per rectam AB eâ-<lb/>dem velocitate. </s>
  <s xml:id="echoid-s8895" xml:space="preserve">At ſæpe non ita facile per horum utrumlibet modum <lb/>_magni@udinum generatio_ declaretur, ſit enim recta AB æquabiliter rc-<lb/>tata (hoc eſt, ita ut temporibus æqualibus æquales efficiat angulos) <lb/>et ſimultaneè punctum M ab A in ipſa recta AB continuo motu <lb/>feratur, etiam uniformi; </s>
  <s xml:id="echoid-s8896" xml:space="preserve">ex iſta _motuum_ compoſitione linea quæ-<lb/>dam producetur, _belix_ ſcilicet _Archimedea_ (nam talia conſultò pro-<lb/>ponimus _exempla, quò celebrium apud Matbematicos magnitudinum_ <lb/>_obiter naturam inſinuem_, et inſtillem minùs ad hæc exercitatis; </s>
  <s xml:id="echoid-s8897" xml:space="preserve">id <lb/>tranſcurrens moneo) cujus generatio per nullos, opinor, mobi-<lb/>
<anchor type="note" xlink:label="note-0203-01a" xlink:href="note-0203-01"/>
lium concurſus, liquidò commodéque ſatis explicetur; </s>
  <s xml:id="echoid-s8898" xml:space="preserve">ita nimirum <lb/>ut motuum iſtorum, vel eorum quantitatem determinantium angulo-<lb/>rum, ſeu linearum, ratio, quantitaſve dignoſcantur. </s>
  <s xml:id="echoid-s8899" xml:space="preserve">Generari qui-<lb/>dem poterit è concurſu paralleli motûs rectæ AC; </s>
  <s xml:id="echoid-s8900" xml:space="preserve">vel circularis <lb/>motûs rectæ BA circa Centrum quodvis B, concurſu cum prædi-<lb/>cto regulari motu circa Centrum A; </s>
  <s xml:id="echoid-s8901" xml:space="preserve">at quæ ſit tum futura recta-<lb/>rum AM, Aμ; </s>
  <s xml:id="echoid-s8902" xml:space="preserve">vel angulorum ABM, ABμ quantitas difficilè <lb/>
<anchor type="note" xlink:label="note-0203-02a" xlink:href="note-0203-02"/>
conſtabit. </s>
  <s xml:id="echoid-s8903" xml:space="preserve">E contrà, ſi recta BA circa Centrum B motu rotetur <lb/>uniformi; </s>
  <s xml:id="echoid-s8904" xml:space="preserve">et ſimul recta AC per AB parallelωs, &amp; </s>
  <s xml:id="echoid-s8905" xml:space="preserve">uniformiter defe-<lb/>ratur, rectarum BA, AC ita latarum interſectio continua lineam quan-<lb/>dam efficiet (illam nempe, quæ quadratrix dici ſolet) cujus ge-<lb/>neratio non ità clarè per ſtrictè dictam motuum compoſitionem ex-<lb/>pediatur, aut explicetùr. </s>
  <s xml:id="echoid-s8906" xml:space="preserve">Generari quidem poteſt per motum re-<lb/>ctum alicujus puncti M in AB delatâ parallelωs ad primò poſitam <lb/>AB; </s>
  <s xml:id="echoid-s8907" xml:space="preserve">vel ex puucto tali in AC parallelo quoque delatâ; </s>
  <s xml:id="echoid-s8908" xml:space="preserve">vel per <lb/>motum puncti in AB, circa B; </s>
  <s xml:id="echoid-s8909" xml:space="preserve">vel circa A rotatâ, rectè ab A <lb/>verſus B, vel à B verſus A decurrentis; </s>
  <s xml:id="echoid-s8910" xml:space="preserve">ſed hujuſmodi ſnppoſi-<lb/>tâ quâpiam motuum compoſitione, quænam ſit rectarum AM, <lb/>aut BM; </s>
  <s xml:id="echoid-s8911" xml:space="preserve">vel angulorum BAM aut ABM aut AMB, vel aliarum <lb/>quarumvis magnitudinum hoſce motus determinantium quantitas, <lb/>aut inter ſe relatio, difficulter innoteſcat. </s>
  <s xml:id="echoid-s8912" xml:space="preserve">Qua præcipuè de cauſa
<pb o="26" file="0204" n="219" rhead=""/>
motuum compoſitionem ab ipſorum concurſu ſecerno; </s>
  <s xml:id="echoid-s8913" xml:space="preserve">quia nem-<lb/>pe magnitudinum generatio nunc uno, nunc alio modò faciliús expli-<lb/>catur. </s>
  <s xml:id="echoid-s8914" xml:space="preserve">Verum ad illos diſtinctius exponendos accedo. </s>
  <s xml:id="echoid-s8915" xml:space="preserve">De compoſitione <lb/>primum. </s>
  <s xml:id="echoid-s8916" xml:space="preserve">Cùm autem motus duobus modis compoſitus intelligi <lb/>poſſit; </s>
  <s xml:id="echoid-s8917" xml:space="preserve">vel nt è pluribus motibus aggregatus, vcl ut de pluribus par-<lb/>ticipans; </s>
  <s xml:id="echoid-s8918" xml:space="preserve">de poſteriore nos diſſertamus; </s>
  <s xml:id="echoid-s8919" xml:space="preserve">quem fortè non meliùs <lb/>quàm prænobilis Philoſophi verbis, &amp; </s>
  <s xml:id="echoid-s8920" xml:space="preserve">exemplis enucleatum dem. <lb/></s>
  <s xml:id="echoid-s8921" xml:space="preserve">
<anchor type="note" xlink:label="note-0204-01a" xlink:href="note-0204-01"/>
“Etſi autem (inquit ille) unumquodque corpus habeat tantùm <lb/>unum motum ſibi proprium, quoniam ab unis tantùm corpori-<lb/>bus ſibi contiguis, et quieſcentibus recedere intelligitur, parti-<lb/>cipare tamen etiam poteſt et de aliis innumeris; </s>
  <s xml:id="echoid-s8922" xml:space="preserve">ſi nempe ſit <lb/>pars aliorum corporum alios motus habentium. </s>
  <s xml:id="echoid-s8923" xml:space="preserve">Ut ſi quis am-<lb/>bulans in navi _horologium in_ pera geſtet, ejus horologii rotu-<lb/>læ unico tantùm motu ſibi proprio movebuntur; </s>
  <s xml:id="echoid-s8924" xml:space="preserve">ſed participa-<lb/>bunt etiam ex alio, quatenus adjunctæ homini ambulanti unam <lb/>cum illo materiæ partem component; </s>
  <s xml:id="echoid-s8925" xml:space="preserve">et ex alio quatenus erunt <lb/>adjunctæ navigio in mari fluctuanti; </s>
  <s xml:id="echoid-s8926" xml:space="preserve">et ex alio quatenus ad-<lb/>junctæ ipſi mari; </s>
  <s xml:id="echoid-s8927" xml:space="preserve">et denique alio, quatenus adjunctæ ipſi terræ, <lb/>ſiquidem tota terra moveatur. </s>
  <s xml:id="echoid-s8928" xml:space="preserve">Omnesque hi motus revera e-<lb/>runt in rotulis iſtis, ſed quia non facilè tam multi ſimul intel-<lb/>ligi, nec etiam omnes agnoſci poſſunt, ſufficiet unicum illum, <lb/>qui proprius eſt cujusque corporis in ipſo conſiderare. </s>
  <s xml:id="echoid-s8929" xml:space="preserve">Ac præ-<lb/>tereà ille unicus cujuſque corporis motus, qui ei proprius eſt, <lb/>inſtar plurium poteſt conſiderari; </s>
  <s xml:id="echoid-s8930" xml:space="preserve">ut cùm in rotis curruum du-<lb/>os diverſos diſtinguimus, unum ſcilicet circa ipſarum axem, et <lb/>alium rectum ſecundum longitudinem viæ per quam feruntur. <lb/></s>
  <s xml:id="echoid-s8931" xml:space="preserve">Sed quòd ideò tales motus non ſint reverà diſtincti patet ex eo, <lb/>quòd unumquodque punctum corporis quod movetur unam tan-<lb/>tùm aliquam lineam deſcribat. </s>
  <s xml:id="echoid-s8932" xml:space="preserve">Nec refert quòd iſta linea ſæpe ſit <lb/>valde contorta, et ideò à pluribus diverſis motibus genita vi-<lb/>deatur, quia poſſumus imaginari eodem modo quamcunque li-<lb/>neam etiam rectam, quæ omnium ſimpliciſſima eſt, ex infini-<lb/>tis diverſis motibus ortam eſſe. </s>
  <s xml:id="echoid-s8933" xml:space="preserve">Ut ſi linea AB feratur verſus <lb/>CD, et eodem tempore punctum A feratur verſus B, linea <lb/>
<anchor type="note" xlink:label="note-0204-02a" xlink:href="note-0204-02"/>
recta AD, quam hoc punctum A deſcribet, non minus pende-<lb/>bit à duobus motibus rectis, abA in B et ab AB in CD, quàm <lb/>linea curva, quæ à quovis rotæ puncto deſcribitur, pendet à <lb/>motu recto et circulari. </s>
  <s xml:id="echoid-s8934" xml:space="preserve">Ac proinde quamvis ſæpe utile ſit u-<lb/>num motum in plures partes hoc pacto diſtinguere ad facilio-<lb/>rem ejus perceptionem; </s>
  <s xml:id="echoid-s8935" xml:space="preserve">abſolutè tamen loquendo unus tantùm
<pb o="27" file="0205" n="220" rhead=""/>
in unoquoque corpore eſt numerandus. </s>
  <s xml:id="echoid-s8936" xml:space="preserve">Ita _Carteſius_. </s>
  <s xml:id="echoid-s8937" xml:space="preserve">Nem-<lb/>pe cùm magnitudo quæpiam exinde quod aliis modo quopiam-<lb/>adnectitur, illorum motus ita particeps eſt, ut ab eo quoad ſi-<lb/>tum ſuum aliquatenus determinetur, iſte motus hujus compoſitio-<lb/>nem quaſi pars ingreditur, ab exemplis poſthac adjungendis res <lb/>luculentius apparebit. </s>
  <s xml:id="echoid-s8938" xml:space="preserve">Motus autem hoc modo componi poſſunt <lb/>_Progreſſivi_ cum _Progreſſivis, Progreſſivi_ cum _Circumlatititis, Cir-_ <lb/>_cumlatitii_ cum _Circumlatitiis_; </s>
  <s xml:id="echoid-s8939" xml:space="preserve">componi poſſunt, inquam, et decom-<lb/>poni modis innumeris; </s>
  <s xml:id="echoid-s8940" xml:space="preserve">quorum omnium cùm inire cenſum im-<lb/>poſſibile ſit, illoſque qui à regularitate deflectunt intelligere difficile <lb/>ſit, exponere difficiliús; </s>
  <s xml:id="echoid-s8941" xml:space="preserve">nos præcipuos ſaltem aliquos, in uſu magìs <lb/>poſitos, et explicatu faciliores attingemus. </s>
  <s xml:id="echoid-s8942" xml:space="preserve">Quales imprimis <lb/>ſunt ii qui è motibus directis et parallelis; </s>
  <s xml:id="echoid-s8943" xml:space="preserve">è directis et rotatitiis, <lb/>è pluribus rotatitiis componuntur; </s>
  <s xml:id="echoid-s8944" xml:space="preserve">præſertim illi quos qui conſti-<lb/>tuunt ſimplices motus omnes vel nonnulli ſunt uniformes. </s>
  <s xml:id="echoid-s8945" xml:space="preserve">Nam <lb/>_uniformitatem nedum R@ſpublic<unsure/>a requirit, ac exigit Eccleſia, ſed_ <lb/>_artes etiam atque ſcientiæ vehementer affectant._ </s>
  <s xml:id="echoid-s8946" xml:space="preserve">Recti motns <lb/>(quibus parallelos à recta linea directos motus adnumero) pri-<lb/>mum ſibi non immeritò locum aſlerunt, ut ſimplicitate præcel-<lb/>lentes, naturæ convenientes et chari, præ cæteris utiles ac uſitati. <lb/></s>
  <s xml:id="echoid-s8947" xml:space="preserve">Nec ulla ſané magnitudinis eſt ſpecies (nulla linea, nulla ſuper-<lb/>ficies, nullum corpus) cujus generatio non è rectis peracta moti-<lb/>bus concipiatur. </s>
  <s xml:id="echoid-s8948" xml:space="preserve">Omnis, inquam, in uno planô conſtituta linea <lb/>procreari poteſt è motu parallelo rectæ lineæ, et puncti in ea; </s>
  <s xml:id="echoid-s8949" xml:space="preserve"><lb/>omnis ſuperficies è motu parallelo plani, et lineæ iu eo (lineæ ſci-<lb/>licet alicujus è rectis modo jam inſinuato motibus progenitæ) <lb/>conſequenter et linea quævis etiam in curva ſuperficie deſignata re-<lb/>ctis motibus effici poteſt. </s>
  <s xml:id="echoid-s8950" xml:space="preserve">Corpus autem ſolidum eodem modo <lb/>genitum intelligatur, quatenus è ſuperficierum genitura reſultat, <lb/>et quatenus ab ipſis ità genitis terminatur, ac circumſcribitur <lb/>Sed quia _ſuperficierum plerarumque curvarum_, quales hactenus _Ma-_ <lb/>_theſis_ excogitavit, &amp; </s>
  <s xml:id="echoid-s8951" xml:space="preserve">linearum in iis non in uno plano jacentium, ge-<lb/>neratio per alios modos commodiùs explicetur, neque mihi quic-<lb/>quam ſuccurrit animadverſione dignum quod de iis dicam, de li-<lb/>nearum ſaltem in uno plano exiſtentium, per rectos et parallelos <lb/>motus generatione diſpiciam. </s>
  <s xml:id="echoid-s8952" xml:space="preserve">Et quidem has quod attinet, earum nul-<lb/>la eſt quæ non ex motu parallelo lineæ rectæ, punctique per e-<lb/>am delati producatur; </s>
  <s xml:id="echoid-s8953" xml:space="preserve">verum hi motus eo contemperari modo de-<lb/>bent, quem ſpecialis lineæ producendæ natura poſcit; </s>
  <s xml:id="echoid-s8954" xml:space="preserve">nec reſert <lb/>qualem, velocitatis reſpectu, motum uni tribuas, ad hujus modò
<pb o="28" file="0206" n="221" rhead=""/>
diverſitatem alterius diverſitas ritè conſequatur accommodeturque. <lb/></s>
  <s xml:id="echoid-s8955" xml:space="preserve">U<unsure/>t e. </s>
  <s xml:id="echoid-s8956" xml:space="preserve">g. </s>
  <s xml:id="echoid-s8957" xml:space="preserve">ſi recta ZA ſemper per rectam AY ſibi parallela feratur <lb/>motu quolibet uniformi, vel difformi (creſcente, vel decreſcente <lb/>vel alternante ſecundum velocitatem, juxta rationem quamvis ima-<lb/>ginabilem) et in ea punctum aliquod M deferatur, ità tamen ut <lb/>puncti motus lineæ rectæ motibus per ſingulas quasque temporis <lb/>partes eaſdem proportionentur, producetur utique linea recta. </s>
  <s xml:id="echoid-s8958" xml:space="preserve">Nem-<lb/>pe ſi fuerit ſemper AB. </s>
  <s xml:id="echoid-s8959" xml:space="preserve">AC:</s>
  <s xml:id="echoid-s8960" xml:space="preserve">: BM. </s>
  <s xml:id="echoid-s8961" xml:space="preserve">Cμ. </s>
  <s xml:id="echoid-s8962" xml:space="preserve">vell<unsure/> AB, MX:</s>
  <s xml:id="echoid-s8963" xml:space="preserve">: AM, <lb/>X μ (poſitâ ſcilicet MX ad AC parallelâ) liquet puncta A, M <lb/>μ in una recta verſari. </s>
  <s xml:id="echoid-s8964" xml:space="preserve">Eſt enim rectæ lineæ proprietas in Ele-<lb/>
<anchor type="note" xlink:label="note-0206-01a" xlink:href="note-0206-01"/>
mento VI. </s>
  <s xml:id="echoid-s8965" xml:space="preserve">demonſtrata, quòd ad eam parallelωs applicatæ rectæ <lb/>lineæ ſuis ad deſignatum in ea punctum diſtantiis proportionales in <lb/>rectam lineam terminantur. </s>
  <s xml:id="echoid-s8966" xml:space="preserve">Quòd ſi motus hi ſic inter ſe contem-<lb/>perentur, ut aſſumptâ quâdam lineâ D habeat rectangulum ex diffe-<lb/>rentia lineæ D, &amp; </s>
  <s xml:id="echoid-s8967" xml:space="preserve">ipſius BM (à puncto mobili decurſæ in recta <lb/>AZ) &amp; </s>
  <s xml:id="echoid-s8968" xml:space="preserve">ipſa BM ad quadratum ex AB (eodem tempore decurſa <lb/>à linea AZ) rationem ſemper eandem progignetur _ellipſis aut cir-_ <lb/>_culus;_ </s>
  <s xml:id="echoid-s8969" xml:space="preserve">circulus quidem ſi ratio propoſita fuerit æqualitas, &amp; </s>
  <s xml:id="echoid-s8970" xml:space="preserve">an-<lb/>gulus ZAY rectus, _ellipſis_ ſi ſecùs; </s>
  <s xml:id="echoid-s8971" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8972" xml:space="preserve">in his erit D una _diame-_ <lb/>_trorum_, ſitum habens in linea AZ primò poſitâ, à vertice A por-<lb/>recta verſus partes Z. </s>
  <s xml:id="echoid-s8973" xml:space="preserve">Sin ità ſe habeant, ut rectangulum ex ſumma <lb/>linearum D, &amp; </s>
  <s xml:id="echoid-s8974" xml:space="preserve">BM &amp; </s>
  <s xml:id="echoid-s8975" xml:space="preserve">ipſa BM ſemper eandem cum quadrato <lb/>e<unsure/>x AB proportionem ſervet, eo compoſito motu procreabitur _by-_ <lb/>_perbole_; </s>
  <s xml:id="echoid-s8976" xml:space="preserve">quadrata quidem illa (vel æquilatera rectangula) ſi _ratio_ <lb/>deſignata fuerit æqualitatis, &amp; </s>
  <s xml:id="echoid-s8977" xml:space="preserve">angulus ZAY rectus; </s>
  <s xml:id="echoid-s8978" xml:space="preserve">ſin aliter, <lb/>alterius, pro rationis aſſignatæ quantitate, ſpeciei; </s>
  <s xml:id="echoid-s8979" xml:space="preserve">cujus _tranſverſa_ <lb/>_diameter_ æquabitur ipſi D, ſitum habens in ZA primò poſita à <lb/>vertice A protenſa verſus partes averſas ab Z; </s>
  <s xml:id="echoid-s8980" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s8981" xml:space="preserve">parameter ex <lb/>ratione data determinatur. </s>
  <s xml:id="echoid-s8982" xml:space="preserve">Quòd ſi perpetuò rectangulum ex ipſa <lb/>D, &amp; </s>
  <s xml:id="echoid-s8983" xml:space="preserve">decurſa BM ad quadratum ex AB eandem perpetuò ra-<lb/>tionem obtinet, conſtabit effici _lineam parabolicam_, cujus _para-_ <lb/>_meter_ ex rectæ D, datæque rationis propoſitæ quantitate facilè <lb/>definietur. </s>
  <s xml:id="echoid-s8984" xml:space="preserve">Et in horum primo quidem caſu ſi motus tranſverſus <lb/>per AY ponatur uniformis, etiam motus deſcendens per AZ unifor-<lb/>mis erit; </s>
  <s xml:id="echoid-s8985" xml:space="preserve">in ſecundo &amp; </s>
  <s xml:id="echoid-s8986" xml:space="preserve">tertio ſi motus per AY ſit uniformis, erit motus <lb/>deſcendens perpetuò creſcens; </s>
  <s xml:id="echoid-s8987" xml:space="preserve">eodemque poſito quoad ultimum caſum, <lb/>in quo parabola fit; </s>
  <s xml:id="echoid-s8988" xml:space="preserve">punctum M continuò velocitate creſcet æqualiter. <lb/></s>
  <s xml:id="echoid-s8989" xml:space="preserve">Nec abſimili modo quævis alia linea tali motûs compoſitione producta <lb/>concipi poteſt. </s>
  <s xml:id="echoid-s8990" xml:space="preserve">Sed ut eò quo tendimus aliquando perveniamus; </s>
  <s xml:id="echoid-s8991" xml:space="preserve"><lb/>agedum videamus ecquid in _rem Mathematicam_ utilitatis ex hujuſ-
<pb o="29" file="0207" n="222" rhead=""/>
modi ſuppoſita linearum generatione poterimus indipiſci. </s>
  <s xml:id="echoid-s8992" xml:space="preserve">Simpli-<lb/>citatis autem &amp; </s>
  <s xml:id="echoid-s8993" xml:space="preserve">perſpicuitatis cauſâ ſupponamus alterum ex his <lb/>motibus, rectæ nimirum paralleliſmum ſervantis, eſſe ſemper uni-<lb/>formem, &amp; </s>
  <s xml:id="echoid-s8994" xml:space="preserve">quænam ex alterius quoad velocitatem generalibus <lb/>differentiis generales emergant linearum productarum affectiones ad-<lb/>nitamur elicere. </s>
  <s xml:id="echoid-s8995" xml:space="preserve">Adnitamur inquam, at proxima lectione.</s>
  <s xml:id="echoid-s8996" xml:space="preserve"/>
</p>
<div xml:id="echoid-div220" type="float" level="2" n="1">
<note position="left" xlink:label="note-0202-01" xlink:href="note-0202-01a" xml:space="preserve">Fig. 13.</note>
<note position="right" xlink:label="note-0203-01" xlink:href="note-0203-01a" xml:space="preserve">Fig. 14.</note>
<note position="right" xlink:label="note-0203-02" xlink:href="note-0203-02a" xml:space="preserve">Fig. 15.</note>
<note position="left" xlink:label="note-0204-01" xlink:href="note-0204-01a" xml:space="preserve">Carteſ.<unsure/>princ. II. <lb/>31, 32.</note>
<note position="left" xlink:label="note-0204-02" xlink:href="note-0204-02a" xml:space="preserve">Fig. 16.</note>
<note position="left" xlink:label="note-0206-01" xlink:href="note-0206-01a" xml:space="preserve">Fig. 17.</note>
</div>
</div>
<div xml:id="echoid-div222" type="section" level="1" n="31">
<head xml:id="echoid-head34" xml:space="preserve"><emph style="sc">Lect</emph>. IV.</head>
<p>
  <s xml:id="echoid-s8997" xml:space="preserve">Propoſitum eſt nobis è compoſitione motuum (qualem proximè <lb/>deſcripſimus) emergentes linearum affectiones indagare ac ex-<lb/>
<anchor type="note" xlink:label="note-0207-01a" xlink:href="note-0207-01"/>
ponere. </s>
  <s xml:id="echoid-s8998" xml:space="preserve">Quorſum imprimìs methodi cauſà repeto ſi recta AZ per <lb/>rectam AY ſibi perpetuò parallela feratur uniformiter, et in ea <lb/>quoque punctum M uniformiter deportetur, quâvis velocitate, li-<lb/>nea recta proveniet. </s>
  <s xml:id="echoid-s8999" xml:space="preserve">Sumantur enim duæ quævis lineæ mobilis <lb/>AZ poſitiones, ad B ſcilicet &amp; </s>
  <s xml:id="echoid-s9000" xml:space="preserve">C; </s>
  <s xml:id="echoid-s9001" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9002" xml:space="preserve">quia motus per AY po-<lb/>nitur uniformis, erunt decurſa ſpatia AB, AC ad ſe, ut _Tempo-_ <lb/>_ra_; </s>
  <s xml:id="echoid-s9003" xml:space="preserve">ſed et ob motum uniformem puncti M etiam rectæ BM, <lb/>CM ſe habebunt ut eadem tempora; </s>
  <s xml:id="echoid-s9004" xml:space="preserve">eſt igitur AB. </s>
  <s xml:id="echoid-s9005" xml:space="preserve">AC:</s>
  <s xml:id="echoid-s9006" xml:space="preserve">: <lb/>BM. </s>
  <s xml:id="echoid-s9007" xml:space="preserve">CM. </s>
  <s xml:id="echoid-s9008" xml:space="preserve">Unde liquet puncta A, M, M in una recta linea ex-<lb/>iſtere. </s>
  <s xml:id="echoid-s9009" xml:space="preserve">Parique ratione conſtat idem de punctis omnibuſcunque, <lb/>quibus punctum M per totum ſuum curſum inſiſtit, aut coincidit, <lb/>Supponatur ſecundò punctum M motu continuo increſcente deſerri <lb/>(juxta quamlibet velocitatis rationem, regulari modo quocunque <lb/>nil intereſt, an irregulari) aio _ſuppoſitionem banc conſectari progeni-_ <lb/>_tarum linearum quas apponemus proprietates generales_ (quales uni <lb/>tali linearum generi convenientes certè præſtat ex unimoda com-<lb/>muni generatione ſimul univerſas elicere, quàm de ſingulis, ut <lb/>
<anchor type="note" xlink:label="note-0207-02a" xlink:href="note-0207-02"/>
paſſim fieri ſolet, ſingulas ſeparatim oſtendere.) </s>
  <s xml:id="echoid-s9010" xml:space="preserve">Notetur inte-<lb/>reà, quòd brevitatis cauſâ motum parallelum uniformem rectæ AZ <lb/>per AY appellabo ſubinde _motum tranſverſum_; </s>
  <s xml:id="echoid-s9011" xml:space="preserve">puncti verò mo-<lb/>ventis ab A in linea AZ motum vocitabo _deſcenſum_, aut _motum_ <lb/>_deſcendentem_, habito ſcilicet ad figuram exhibitam reſpectu. </s>
  <s xml:id="echoid-s9012" xml:space="preserve">Item <lb/>quòd, ob motûs per AY et ei parallelas uniformitatem, poſſit <lb/>ea cum ipſius partibus motûs tempus, et ejus partes repræſentare. <lb/></s>
  <s xml:id="echoid-s9013" xml:space="preserve">Jam ad dictas proprietates expendendas accedo.</s>
  <s xml:id="echoid-s9014" xml:space="preserve"/>
</p>
<div xml:id="echoid-div222" type="float" level="2" n="1">
<note position="right" xlink:label="note-0207-01" xlink:href="note-0207-01a" xml:space="preserve">Fig. 18.</note>
<note position="right" xlink:label="note-0207-02" xlink:href="note-0207-02a" xml:space="preserve">Fig. 19.</note>
</div>
<pb o="30" file="0208" n="223" rhead=""/>
<p>
  <s xml:id="echoid-s9015" xml:space="preserve">I. </s>
  <s xml:id="echoid-s9016" xml:space="preserve">Hoc modo (per motum nempe tranſverſum uniformem, &amp; </s>
  <s xml:id="echoid-s9017" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0208-01a" xlink:href="note-0208-01"/>
deſcenſivum continuo creſcentem) progenita linea per omnes ſui partes <lb/>curva evadet.</s>
  <s xml:id="echoid-s9018" xml:space="preserve">‖ Accipiantur enim in ipſa tria quælibet puncta M, <lb/>N, O; </s>
  <s xml:id="echoid-s9019" xml:space="preserve">per quæ tranſeant BZ, CZ, DZ ad AZ parallelæ, &amp; </s>
  <s xml:id="echoid-s9020" xml:space="preserve">per <lb/>puncta M, N ducatur recta MNK. </s>
  <s xml:id="echoid-s9021" xml:space="preserve">Et quia recta MN gignitur è <lb/>motu compoſito tranſverſo per BC (vel huic parallelam MG) &amp; </s>
  <s xml:id="echoid-s9022" xml:space="preserve"><lb/>deſcendente per AZ, uniformi utroque; </s>
  <s xml:id="echoid-s9023" xml:space="preserve">tranſverſus autem per MG <lb/>eſt prorſus idem cum tranſverſo, quo linea propoſita MNO de-<lb/>ſcribitur; </s>
  <s xml:id="echoid-s9024" xml:space="preserve">patet velocitatem deſcendentis motûs uniformis rectam MN <lb/>gignentis minorem eſſe velocitate, quam motus itidem deſcendens, <lb/>lineam MNO deſcribens, habet in N (etenim niſi motus hic velo-<lb/>cior jam ſit illo, cùm continuò creſcere ponatur, in toto tempore <lb/>deſcenſus per GN illo tardior fuiſſet, adeóque nunquam eodem tem-<lb/>pore ſpatium æquale tranſegiſſet, nec unà cum eo pertigiſſet ad <lb/>punctum N) ergò motus hîc inæqualis &amp; </s>
  <s xml:id="echoid-s9025" xml:space="preserve">increſcens per tempus <lb/>motûs uniformis CD continuatus (quo nempe gignitur linea NO) <lb/>majus ſpatium emetitur, quàm uniformis motus deſcendens, quo <lb/>MN. </s>
  <s xml:id="echoid-s9026" xml:space="preserve">ad K protractus deſcribitur, eodem tempore CD; </s>
  <s xml:id="echoid-s9027" xml:space="preserve">(liquet <lb/>enim eodem tempore à majore vi creſcente majus ſpatium peragi, quàm <lb/>à minore neutiquam creſcente) quare linea HO major eſt quàm HK; <lb/></s>
  <s xml:id="echoid-s9028" xml:space="preserve">adeóque tria puncta M, N, O non exiſtunt in eadem recta linea; </s>
  <s xml:id="echoid-s9029" xml:space="preserve"><lb/>quod cùm tribus quibuſvis lineæ MNO punctis conveniat, abunde <lb/>patet eam eſſe nullibi rectam, ſed per omnes ſui partes incurvatam, &amp; </s>
  <s xml:id="echoid-s9030" xml:space="preserve"><lb/>inflexam.</s>
  <s xml:id="echoid-s9031" xml:space="preserve"/>
</p>
<div xml:id="echoid-div223" type="float" level="2" n="2">
<note position="left" xlink:label="note-0208-01" xlink:href="note-0208-01a" xml:space="preserve">Fig. 19.</note>
</div>
<p>
  <s xml:id="echoid-s9032" xml:space="preserve">II. </s>
  <s xml:id="echoid-s9033" xml:space="preserve">Hinc emergit _Corollarium_; </s>
  <s xml:id="echoid-s9034" xml:space="preserve">velocitas motûs uniformis deſcen-<lb/>dentis, quo curvæ MNO ſubtenſa quævis (ut MN) deſcribitur, <lb/>exiſtente ſcilicet communi tranſverſo motu uniformi quo ipſa, ejúſque <lb/>arcus fiunt, minor eſt velocitate, quàm motus deſcenſivus increſcens <lb/>habet ad communem utriuſque terminum N.</s>
  <s xml:id="echoid-s9035" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9036" xml:space="preserve">III. </s>
  <s xml:id="echoid-s9037" xml:space="preserve">Hujuſce curvæ _ſubtenſa_ quælibet (ut MO) intra _ſuum arcum_ <lb/>(verſus partes AZ) tota cadit, &amp; </s>
  <s xml:id="echoid-s9038" xml:space="preserve">producta tota cadit extra lineam <lb/>MNO.</s>
  <s xml:id="echoid-s9039" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9040" xml:space="preserve">Nam ſi ſumatur in arcu MO punctum quodvis N, &amp; </s>
  <s xml:id="echoid-s9041" xml:space="preserve">connectantur <lb/>
<anchor type="note" xlink:label="note-0208-02a" xlink:href="note-0208-02"/>
rectæ MN, NO liquet totam MO intra rectas MN, NO jacere, <lb/>&amp; </s>
  <s xml:id="echoid-s9042" xml:space="preserve">proinde intra curvam MNO. </s>
  <s xml:id="echoid-s9043" xml:space="preserve">Tota verò, ſi producatur, extra <lb/>lineam MNO cadit, quia nuſquam alibi ei occurrit, utì mox oſtenſum.</s>
  <s xml:id="echoid-s9044" xml:space="preserve">‖</s>
</p>
<div xml:id="echoid-div224" type="float" level="2" n="3">
<note position="left" xlink:label="note-0208-02" xlink:href="note-0208-02a" xml:space="preserve">Elem. III. 2. <lb/>Apoll. I. <lb/>Seren. I. 8.</note>
</div>
<p>
  <s xml:id="echoid-s9045" xml:space="preserve">Hoc accidens de circulo ſpeciatim demonſtrat _Euclides, de ſectionibus_ <lb/>conicis Apollenius; </s>
  <s xml:id="echoid-s9046" xml:space="preserve">de cylindricis Serenus<unsure/>.</s>
  <s xml:id="echoid-s9047" xml:space="preserve"/>
</p>
<pb o="31" file="0209" n="224" rhead=""/>
<p>
  <s xml:id="echoid-s9048" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s9049" xml:space="preserve">Patet curvam propoſitam eſſe convexam, aut concavam ad <lb/>eaſdem partes (convexam verſus partes ſuperiores vel exteriores AY, <lb/>concavam introrſum, aut deorſum verſus AZZ) nam hoc ipſum, <lb/>fore convexum aut concavum ad eaſdem partes, nil omnino deſignat <lb/>aliud, quàm à nulla recta linea præterquam duobus punctis ſecari; <lb/></s>
  <s xml:id="echoid-s9050" xml:space="preserve">nec aliò recidit, quam initio libri de ſphæra &amp; </s>
  <s xml:id="echoid-s9051" xml:space="preserve">cylindro tradit _Ar-_ <lb/>_chimedes,_ lineæ ad eaſdem partes cavæ definitio. </s>
  <s xml:id="echoid-s9052" xml:space="preserve">Perſpicuum eſt <lb/>v. </s>
  <s xml:id="echoid-s9053" xml:space="preserve">g. </s>
  <s xml:id="echoid-s9054" xml:space="preserve">ut linea MN duobus in punctis M, N curvam MNO ſecans ei <lb/>rurſus occurrat, ut puta in K, debere curvam MNO reflecti, ver-<lb/>ſùſque partes AY recurvari; </s>
  <s xml:id="echoid-s9055" xml:space="preserve">id quod modò demonſtratum eſt non <lb/>poſſe contingere. </s>
  <s xml:id="echoid-s9056" xml:space="preserve">Quapropter ipſa linea verſus eaſdem partes con-<lb/>vexa eſt, ſeu concava.</s>
  <s xml:id="echoid-s9057" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9058" xml:space="preserve">V. </s>
  <s xml:id="echoid-s9059" xml:space="preserve">Apertiſſimè conſtat lineas quaſvis rectas (ut BZ, CZ) gene-<lb/>trici AZ parallelas propoſitam curvam ſecare (modò contineantur <lb/>intra terminos motûs per AY; </s>
  <s xml:id="echoid-s9060" xml:space="preserve">quia curva per harum quamvis inde-<lb/>finitè promotam deſcripta cenſetur) addo quod harum quælibet cur-<lb/>vam in uno tantùm puncto ſecat.</s>
  <s xml:id="echoid-s9061" xml:space="preserve">‖ Id patet, quia recta genetrix <lb/>AZ per unicum duntaxat inſtans temporis durat in ſitu quovis uno, <lb/>ſeu BZ; </s>
  <s xml:id="echoid-s9062" xml:space="preserve">ſimúlque pertingit ipſam BZ, ac deſerit; </s>
  <s xml:id="echoid-s9063" xml:space="preserve">prætérque <lb/>punctum unum M in BMZ reliqua cuncta lineæ curvæ puncta ſunt <lb/>
<anchor type="note" xlink:label="note-0209-01a" xlink:href="note-0209-01"/>
in parallelis ad BZ. </s>
  <s xml:id="echoid-s9064" xml:space="preserve">Ergò liquidum eſt ipſam BZ in uno tantùm <lb/>puncto curvam ſecare.</s>
  <s xml:id="echoid-s9065" xml:space="preserve">‖ Hocipſum de parabola, &amp; </s>
  <s xml:id="echoid-s9066" xml:space="preserve">hiperbola ſpecia-<lb/>tim oſtendit _Apollonius_; </s>
  <s xml:id="echoid-s9067" xml:space="preserve">de ſectionibus conoideon _Arcbimedes_.</s>
  <s xml:id="echoid-s9068" xml:space="preserve"/>
</p>
<div xml:id="echoid-div225" type="float" level="2" n="4">
<note position="right" xlink:label="note-0209-01" xlink:href="note-0209-01a" xml:space="preserve">Apoll. I. 26. <lb/>Arch. de Con<unsure/>@id. <lb/>&amp; Sph. 16.</note>
</div>
<p>
  <s xml:id="echoid-s9069" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s9070" xml:space="preserve">Non diſſimili modo patet ad AY parallelam quamvis, (qualis <lb/>PG) unico puncto propoſitam curvam attingere.</s>
  <s xml:id="echoid-s9071" xml:space="preserve">‖ Quòd ſemel <lb/>occurret (modò contineatur intra limites deſcensûs per AZ) patet, <lb/>quia punctum mobile continuò deſcendens, indefinito progreſſu, eam <lb/>indefinitè protenſam aliquando trajiciet; </s>
  <s xml:id="echoid-s9072" xml:space="preserve">nec in eo tamen præterquam <lb/>
<anchor type="note" xlink:label="note-0209-02a" xlink:href="note-0209-02"/>
ad unum temporis momentum perdurat.</s>
  <s xml:id="echoid-s9073" xml:space="preserve">‖ Videatur hoc de ſectioni-<lb/>bus conicis oſtendens _Apollonius_.</s>
  <s xml:id="echoid-s9074" xml:space="preserve"/>
</p>
<div xml:id="echoid-div226" type="float" level="2" n="5">
<note position="right" xlink:label="note-0209-02" xlink:href="note-0209-02a" xml:space="preserve">I. 19.</note>
</div>
<p>
  <s xml:id="echoid-s9075" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s9076" xml:space="preserve">Patet omnes curvæ ſubtenſas rectas cum AZ &amp; </s>
  <s xml:id="echoid-s9077" xml:space="preserve">ei parallelis, <lb/>ſi producantur, concurrere.</s>
  <s xml:id="echoid-s9078" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9079" xml:space="preserve">Quòd enim ſubtenſa quævis, ut MN, uni parallelarum alicui, ut <lb/>BR, occurrit, ibi ſcilicet ubi ipſa curvam ſecat, exinde manifeſtiſſimum <lb/>eſt, quòd tota curva per parallelum dictæ rectæ motum deſcribitur. <lb/></s>
  <s xml:id="echoid-s9080" xml:space="preserve">Ergò, cùm uni occurrat, omnibus occurret; </s>
  <s xml:id="echoid-s9081" xml:space="preserve">quæ enim uni paralle-
<pb o="32" file="0210" n="225" rhead=""/>
larum æquidiſtat recta, pariter omnibus æquidiſtat, ut in elemento <lb/>primo demonſtratur.</s>
  <s xml:id="echoid-s9082" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9083" xml:space="preserve">Operæ pretium exiſtimavit _Apollonius_ hoc de _parabola, &amp; </s>
  <s xml:id="echoid-s9084" xml:space="preserve">byper-_ <lb/>
<anchor type="note" xlink:label="note-0210-01a" xlink:href="note-0210-01"/>
_bola_ ſpeciatim demonſtrare.</s>
  <s xml:id="echoid-s9085" xml:space="preserve"/>
</p>
<div xml:id="echoid-div227" type="float" level="2" n="6">
<note position="left" xlink:label="note-0210-01" xlink:href="note-0210-01a" xml:space="preserve">I. 22.</note>
</div>
<p>
  <s xml:id="echoid-s9086" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s9087" xml:space="preserve">Simili modo patet rectas quaſcunque curvas tangentes una <lb/>tantùm excipitur, ad extremum lineæ recurrentis. </s>
  <s xml:id="echoid-s9088" xml:space="preserve">Vid. </s>
  <s xml:id="echoid-s9089" xml:space="preserve">18. </s>
  <s xml:id="echoid-s9090" xml:space="preserve">hujus. <lb/></s>
  <s xml:id="echoid-s9091" xml:space="preserve">Iiſdem parallelis occurrere.</s>
  <s xml:id="echoid-s9092" xml:space="preserve">‖ Etiam hoc, quoad _ſectiones conicas_, uno <lb/>
<anchor type="note" xlink:label="note-0210-02a" xlink:href="note-0210-02"/>
vel altero _Theoremate_ demonſtravit _Apollonius_.</s>
  <s xml:id="echoid-s9093" xml:space="preserve"/>
</p>
<div xml:id="echoid-div228" type="float" level="2" n="7">
<note position="left" xlink:label="note-0210-02" xlink:href="note-0210-02a" xml:space="preserve">I. 24, 25.</note>
</div>
<p>
  <s xml:id="echoid-s9094" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s9095" xml:space="preserve">Quinimò rectæ quævis ipſam AZ ſecantes (infra punctum <lb/>A, ſupráque limitem, ſiquis erit, motûs deſcenſivi) curvam <lb/>ſecabunt.</s>
  <s xml:id="echoid-s9096" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9097" xml:space="preserve">Cùm enim omnes ipſi AZ parallelas ſecent etiam infinitè pro-<lb/>ductæ curvam ſecent oportet. </s>
  <s xml:id="echoid-s9098" xml:space="preserve">_Hujuſmodi Symptomatis demonſtra-_ <lb/>
<anchor type="note" xlink:label="note-0210-03a" xlink:href="note-0210-03"/>
_tioni in ſectionibus conicis_ laborioſam operam impendit _Apolloniu_.</s>
  <s xml:id="echoid-s9099" xml:space="preserve"/>
</p>
<div xml:id="echoid-div229" type="float" level="2" n="8">
<note position="left" xlink:label="note-0210-03" xlink:href="note-0210-03a" xml:space="preserve">I. 27, 28.</note>
</div>
<p>
  <s xml:id="echoid-s9100" xml:space="preserve">X. </s>
  <s xml:id="echoid-s9101" xml:space="preserve">Porrò liquet applicatas ad rectam AY, ipſi AZ parallelas <lb/>(quas nempe propoſitæ curvæ ſinus verſos appellare fas erit mi-<lb/>norem inter ſe rationem habere (minores cum majoribus comparan-<lb/>do, ſeu minores antecedentium loco ponendo) quàm habent re-<lb/>ſpectivæ AY partes, iiſdem temporibus decurſæ (quas &amp; </s>
  <s xml:id="echoid-s9102" xml:space="preserve">curvæ <lb/>propoſitæ ſinus rectos appellare nil dubitem.) </s>
  <s xml:id="echoid-s9103" xml:space="preserve">Nempe BM ad CN <lb/>minorem rationem habet, quàm AB ad AC, vel BM ad CF; </s>
  <s xml:id="echoid-s9104" xml:space="preserve">quia <lb/>CN &amp;</s>
  <s xml:id="echoid-s9105" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s9106" xml:space="preserve">CF.</s>
  <s xml:id="echoid-s9107" xml:space="preserve">‖ Hoc de circulis, &amp; </s>
  <s xml:id="echoid-s9108" xml:space="preserve">aliis curvis ſpeciatim reperiatur <lb/>paſſim oſtenſum.</s>
  <s xml:id="echoid-s9109" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9110" xml:space="preserve">Ad ſequentia notandum, quod ſi recta tranſverſim &amp; </s>
  <s xml:id="echoid-s9111" xml:space="preserve">parallelωs <lb/>mota retrogradè (à D puta verſus A per DA) moveri concipiatur, <lb/>ab aliquo curvæ propoſitæ puncto, velut O, incipiens; </s>
  <s xml:id="echoid-s9112" xml:space="preserve">eâdemque ſem-<lb/>per ratione dictum punctum ab O aſcendens quoad velocitatem de-<lb/>creſcat, quâ ad ipſum O deſcendens increverat, eadem curva pro-<lb/>ducetur. </s>
  <s xml:id="echoid-s9113" xml:space="preserve">Quidni? </s>
  <s xml:id="echoid-s9114" xml:space="preserve">Cùm idem motus ſit, inversè tantum conſide-<lb/>ratus.</s>
  <s xml:id="echoid-s9115" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9116" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s9117" xml:space="preserve">Supponatur rectam lineam TMS propoſitam curvam in <lb/>puncto M tangere (ſic ut eam nempe non ſecet) occurrátque tangens <lb/>hæc rectæ AZ in T, ducatúrque per M recta PMG ad AY parallela; <lb/></s>
  <s xml:id="echoid-s9118" xml:space="preserve">dico velocitatem puncti deſcendentis, e<unsure/>óque motu curvam deſcriben-<lb/>tis, quam habet ad contactum M, æquari velocitati, quâ recta <lb/>TP deſcribetur uniformiter eodem tempore, quo recta AZ fertur
<pb o="33" file="0211" n="226" rhead=""/>
per AC vel PM. </s>
  <s xml:id="echoid-s9119" xml:space="preserve">(vel, quòd eodem recidit, dico quòd velocitas <lb/>puncti deſcendentis in M ad velocitatem quâ fertur recta AZ ſe <lb/>habet, ut recta TP ad PM.) </s>
  <s xml:id="echoid-s9120" xml:space="preserve">Sumatur enim ubivis in tangente <lb/>punctum aliquod K, &amp; </s>
  <s xml:id="echoid-s9121" xml:space="preserve">per ipſum ducatur recta KG, curvæ occur-<lb/>rens in O, parallelis autem AY, &amp; </s>
  <s xml:id="echoid-s9122" xml:space="preserve">PG in D, &amp; </s>
  <s xml:id="echoid-s9123" xml:space="preserve">G. </s>
  <s xml:id="echoid-s9124" xml:space="preserve">Et quia <lb/>tangens TM duplici concipiatur uniformi motu deſcripta, altero <lb/>rectæ TZ per AC vel PM parallelωs delatæ, altero puncti deſcen-<lb/>dentis à T per TZ; </s>
  <s xml:id="echoid-s9125" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9126" xml:space="preserve">ſit horum motuum alter per AC, vel <lb/>PM communis vel idem cum illo quo curva deſcribitnr; </s>
  <s xml:id="echoid-s9127" xml:space="preserve">cùm TZ <lb/>eſt in ſitu KG, erit AZ in eodem; </s>
  <s xml:id="echoid-s9128" xml:space="preserve">ergò cùm punctum à T deſcendens <lb/>fuerit in K, erit punctum ab A deſcendens in curvæ cum KG in-<lb/>terſectione O (nec enim, ut anteà deductum eſt, alibi recta KG <lb/>curvam ſecat) eſt autem punctum O infra K quia tangens extra cur-<lb/>vam tota verſatur. </s>
  <s xml:id="echoid-s9129" xml:space="preserve">Jam ſi punctum K ponatur ſupra contactum <lb/>verſus T, quoniam tum OG minor eſt quàm KG, liquet velo-<lb/>citatem puncti deſcendentis, quo curva deſcribitur, in curvæ pun-<lb/>cto O minorem eſſe velocitate motûs uniformis deſcendentis, quâ <lb/>tangens efficitur; </s>
  <s xml:id="echoid-s9130" xml:space="preserve">quoniam illa ſemper increſcens eodem tempore <lb/>(per GM repræſentato) minus ſpatium tranſigit, quàm hæc mi-<lb/>nimè creſcens; </s>
  <s xml:id="echoid-s9131" xml:space="preserve">aſt eadem continuo perſeverans; </s>
  <s xml:id="echoid-s9132" xml:space="preserve">illa ſcilicet rectam <lb/>OG hæc rectam KG conficit. </s>
  <s xml:id="echoid-s9133" xml:space="preserve">Contra vero ſi punctum K infra <lb/>contactum ad partes S exiſtat, quoniam OG tum major eſt quàm <lb/>KG, patet velocitatem puncti deſcendentis, quo curva fit, in pun-<lb/>cto O majorem eſſe velocitate motûs uniformis itidem deſcenden-<lb/>tis, quo tangens efficitur; </s>
  <s xml:id="echoid-s9134" xml:space="preserve">quia motus iſte, continuò decreſcens <lb/>eodem per GM tempore, majus peragit ſpatium OG, quàm hic <lb/>
<anchor type="note" xlink:label="note-0211-01a" xlink:href="note-0211-01"/>
minimè decreſcens, at in eodem tenore perſiſtens, conficit, ip-<lb/>ſum nempe ſpatium KG. </s>
  <s xml:id="echoid-s9135" xml:space="preserve">Ergo cùm velocitas curvam deſcribentis <lb/>puncti quovis in curvæ puncto ſupra contactum verſus A minor ſit <lb/>velocitate motûs per TP; </s>
  <s xml:id="echoid-s9136" xml:space="preserve">quovis autem in puncto infra contactum <lb/>eâdem major; </s>
  <s xml:id="echoid-s9137" xml:space="preserve">liquet in ipſo contactu M ei penitus exæquari. <lb/></s>
  <s xml:id="echoid-s9138" xml:space="preserve">Q.</s>
  <s xml:id="echoid-s9139" xml:space="preserve">E.</s>
  <s xml:id="echoid-s9140" xml:space="preserve">D.</s>
  <s xml:id="echoid-s9141" xml:space="preserve"/>
</p>
<div xml:id="echoid-div230" type="float" level="2" n="9">
<note position="right" xlink:label="note-0211-01" xlink:href="note-0211-01a" xml:space="preserve">Fig. 20.</note>
</div>
<p>
  <s xml:id="echoid-s9142" xml:space="preserve">XII Hujus converſa, conſimili diſcurſu, rem breviùs exponendo, <lb/>demonſtretur. </s>
  <s xml:id="echoid-s9143" xml:space="preserve">Nempe, ſi velocitas puncti deſcendentis ab A in a-<lb/>liquo curvæ puncto M æquetur velocitati, quâ punctum T uni-<lb/>formiter latum, rectam TP deſcriberet tempore PM vel AC <lb/>(vel ſit velocitas motûs deſcendentis ad M ad velocitatem motûs <lb/>tranſverſi, ut TP ad PM) recta TMS curvam AMO tan-<lb/>get ad M.</s>
  <s xml:id="echoid-s9144" xml:space="preserve"/>
</p>
<pb o="34" file="0212" n="227" rhead=""/>
<p>
  <s xml:id="echoid-s9145" xml:space="preserve">Nam ſumpto quovis in recta TS puncto K, &amp; </s>
  <s xml:id="echoid-s9146" xml:space="preserve">ductâ KG ad <lb/>AZ parallelâ; </s>
  <s xml:id="echoid-s9147" xml:space="preserve">quoniam verſus partes AT velocitas aſcendentis <lb/>puncti, curvam efficientis, ſemper decreſcit ab M ad O, illi verò <lb/>ex hypotheſi par velocitas puncti rectam MT gignentis haud de-<lb/>creſcit ab M ad K, ſitque tempus MG commune, erit ſpatium <lb/>GO minus quàm GK; </s>
  <s xml:id="echoid-s9148" xml:space="preserve">unde punctum K erit extra curvam. </s>
  <s xml:id="echoid-s9149" xml:space="preserve">Item, <lb/>quia verſus alteras partes, velocitas deſcendentis, quo curva fit, in-<lb/>creſcit ſemper ab M verſus O; </s>
  <s xml:id="echoid-s9150" xml:space="preserve">æqualis autem ei velocitas, quâ recta <lb/>MS fit, haud creſcit ab M ad K; </s>
  <s xml:id="echoid-s9151" xml:space="preserve">idémque ſit rurſus tempus MG, <lb/>liquet rectam GO excedere rectam GK; </s>
  <s xml:id="echoid-s9152" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9153" xml:space="preserve">idc<unsure/>irco punctum K ſupra <lb/>curvam exiſtere. </s>
  <s xml:id="echoid-s9154" xml:space="preserve">Quare mani@eſtum eſt omnia dictæ rectæ puncta <lb/>extra curvam exiſtere; </s>
  <s xml:id="echoid-s9155" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9156" xml:space="preserve">eam proinde curvam contingere: <lb/></s>
  <s xml:id="echoid-s9157" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s9158" xml:space="preserve">E. </s>
  <s xml:id="echoid-s9159" xml:space="preserve">D.</s>
  <s xml:id="echoid-s9160" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9161" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s9162" xml:space="preserve">Ex hiſce ſtatim _conſectatur, hujuſmodi curvas ad unum_ <lb/>_punctum ab una tantùm recta contingi._</s>
  <s xml:id="echoid-s9163" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9164" xml:space="preserve">Nam tangere ponatur recta MT curvam AMO ad M; </s>
  <s xml:id="echoid-s9165" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9166" xml:space="preserve">ſi <lb/>fieri poteſt altera MX etiam tangat. </s>
  <s xml:id="echoid-s9167" xml:space="preserve">Ergo eodem tempore, eâdem <lb/>velocitate (illâ ſcilicet, quæ puncti curvam deſcribentis ad contactum <lb/>M acquiſitæ velocitati æquatur) deſcribetur utraque recta XP, TM; <lb/></s>
  <s xml:id="echoid-s9168" xml:space="preserve">quare XP, TP æquales eru<unsure/>nt, totum &amp; </s>
  <s xml:id="echoid-s9169" xml:space="preserve">pars: </s>
  <s xml:id="echoid-s9170" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s9171" xml:space="preserve">E. </s>
  <s xml:id="echoid-s9172" xml:space="preserve">A. </s>
  <s xml:id="echoid-s9173" xml:space="preserve">Ergo <lb/>non tanget altera præter poſitam MT.</s>
  <s xml:id="echoid-s9174" xml:space="preserve">‖ _Hanc ſpeciatim de circule_ <lb/>_demonſtravit Euclides; </s>
  <s xml:id="echoid-s9175" xml:space="preserve">de Sectionibus Conicis Apollonius_, de lineis <lb/>aliis alii. </s>
  <s xml:id="echoid-s9176" xml:space="preserve">Exhinc _Lucrum_ emergit haud aſpernandum, quòd eâdem <lb/>operâ _propoſitiones de tangentibus inve ſæ demonſtrantur._ </s>
  <s xml:id="echoid-s9177" xml:space="preserve">Nempe ſi <lb/>determinetur angulus PMT (vel alter quiſpiam quem recta po-<lb/>ſitione data cum tangente facit ad punctum curvæ deſignatum) aut ſi <lb/>determinetur quantitas rectæ PT (vel ſimilis cujuſpiam alterius à <lb/>
<anchor type="note" xlink:label="note-0212-01a" xlink:href="note-0212-01"/>
puncto in data poſitione recta deſignato per tangentem interceptæ) <lb/>eo tangens determinabitur. </s>
  <s xml:id="echoid-s9178" xml:space="preserve">Et permutatim, ſi tangens ſitu deter-<lb/>
<anchor type="note" xlink:label="note-0212-02a" xlink:href="note-0212-02"/>
minetur, angulorum atque linearum ejuſmodi quantitas indè digno-<lb/>ſcetur. </s>
  <s xml:id="echoid-s9179" xml:space="preserve">Adeóque parcetur operæ, qualem inſumpſerunt plerique <lb/>tales propoſitiones inverſas demonſtrandi. </s>
  <s xml:id="echoid-s9180" xml:space="preserve">Quod &amp; </s>
  <s xml:id="echoid-s9181" xml:space="preserve">eo magìs ob-<lb/>ſervatu dignum eſt, quia ſæpe talium inverſarum propoſitionum <lb/>una quàm altera longè promptiùs invenitur, atque faciliùs demon-<lb/>ſtratur. </s>
  <s xml:id="echoid-s9182" xml:space="preserve">Cujus obſervationis, niſi longiùs evagari nollem, in promptu <lb/>forent _Specimina_.</s>
  <s xml:id="echoid-s9183" xml:space="preserve"/>
</p>
<div xml:id="echoid-div231" type="float" level="2" n="10">
<note position="left" xlink:label="note-0212-01" xlink:href="note-0212-01a" xml:space="preserve">_Eucl. III._ 16, <lb/>17.</note>
<note position="left" xlink:label="note-0212-02" xlink:href="note-0212-02a" xml:space="preserve">_Apoll. I._ 32, 33, <lb/>34, 35, 36.</note>
</div>
<p>
  <s xml:id="echoid-s9184" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s9185" xml:space="preserve">E dictis infertur puncti deſcendentis velocitates in duobus <lb/>quibuſvis deſignatis curvæ punctis ad ſe proportionem habere reciprocè
<pb o="35" file="0213" n="228" rhead=""/>
compoſitam è rationibus applicatarum ab iſtis punctis ad rectam AZ <lb/>(ipſi ſcilicet AY parallelarum) &amp; </s>
  <s xml:id="echoid-s9186" xml:space="preserve">interceptarum à tangentibus ad iſta <lb/>puncta ac dictis applicatis (vel, rationem velocitatum æquari rationi <lb/>applicatarum ex interceptarum ratione ſubductæ.)</s>
  <s xml:id="echoid-s9187" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9188" xml:space="preserve">Nempe ſi duæ rectæ MT, NX curvam tangent ad puncta M, N; <lb/></s>
  <s xml:id="echoid-s9189" xml:space="preserve">protractæ ZA occurrentes in T, X; </s>
  <s xml:id="echoid-s9190" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9191" xml:space="preserve">applicentur NP, NQ ad <lb/>YA parallelæ, velocitatum ad puncta, M, N proportio componetur <lb/>è proportione ipſius TP ad PM, &amp; </s>
  <s xml:id="echoid-s9192" xml:space="preserve">ipſius QN ad QX. </s>
  <s xml:id="echoid-s9193" xml:space="preserve">Nam <lb/>
<anchor type="note" xlink:label="note-0213-01a" xlink:href="note-0213-01"/>
velocitas in M ad velocitatem uniformem per AY ſe habet ut TP ad <lb/>PM; </s>
  <s xml:id="echoid-s9194" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9195" xml:space="preserve">velocitas iſta uniformis ſe habet ad velocitatem in N, ut <lb/>QN ad QX. </s>
  <s xml:id="echoid-s9196" xml:space="preserve">Ergo velocitas in M ad velocitatem in N ex his <lb/>duabus rationibus PP ad PM, &amp; </s>
  <s xml:id="echoid-s9197" xml:space="preserve">QN ad QX componetur Notetur à <lb/>concurſu tangentium ductâ FE ad AY parallelâ; </s>
  <s xml:id="echoid-s9198" xml:space="preserve">fore TE, XE <lb/>= TP. </s>
  <s xml:id="echoid-s9199" xml:space="preserve">PM + QN. </s>
  <s xml:id="echoid-s9200" xml:space="preserve">QX.</s>
  <s xml:id="echoid-s9201" xml:space="preserve"/>
</p>
<div xml:id="echoid-div232" type="float" level="2" n="11">
<note position="right" xlink:label="note-0213-01" xlink:href="note-0213-01a" xml:space="preserve">Fig. 21.</note>
</div>
<p>
  <s xml:id="echoid-s9202" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s9203" xml:space="preserve">Obiter interjicio generalem hinc &amp; </s>
  <s xml:id="echoid-s9204" xml:space="preserve">bene facilem conſequi <lb/>_Problematis iſtius ſolutionem_, quam tanti fecit, &amp; </s>
  <s xml:id="echoid-s9205" xml:space="preserve">cui tantum laborem <lb/>impendit G_alilæus_, quámque _Torricellius_ pronunciat eum quàm optimè <lb/>&amp; </s>
  <s xml:id="echoid-s9206" xml:space="preserve">ingenioſiſſimè reperiſſe. </s>
  <s xml:id="echoid-s9207" xml:space="preserve">Rem ità proponit _Torricellius_ (nam ipſe <lb/>_Galilæus_ ad manum non eſt) propoſitâ quâvis _parabolâ_, cujus <lb/>_vertex_ A oportet punctum aliquod ſublime reperire; </s>
  <s xml:id="echoid-s9208" xml:space="preserve">è quo ſi grave <lb/>
<anchor type="note" xlink:label="note-0213-02a" xlink:href="note-0213-02"/>
cadat uſque ad A, &amp; </s>
  <s xml:id="echoid-s9209" xml:space="preserve">ex puncto cum impetu jam concepto horizonta-<lb/>liter convertatur, ipſa _propoſitam parabolam_ deſcribat (notetur, quod <lb/>motus deſcenſivus parabolam deſcribens non è puncto ſublimi, ſed ab <lb/>ipſo puncto A cenſetur inchoare.) </s>
  <s xml:id="echoid-s9210" xml:space="preserve">Huc recidit _Problema, @ alilæi_ ſup-<lb/>poſitis inſiſtendo, ut determinentur particulares velocitates motuum, <lb/>uniformis horizontalis, ſeu tranſverſi, &amp; </s>
  <s xml:id="echoid-s9211" xml:space="preserve">æqualiter creſcentis deſcen-<lb/>ſivi quorum è compoſitione deſcripta concipitur exhibita parabola. <lb/></s>
  <s xml:id="echoid-s9212" xml:space="preserve">Nos illud, quæcunque ſit creſcentis deſcenſivi motûs ratio, quicunque <lb/>modus, generaliter exequemur; </s>
  <s xml:id="echoid-s9213" xml:space="preserve">ſpecialem illum de _parobola_ caſum in <lb/>exemplum ſubjuncturi.</s>
  <s xml:id="echoid-s9214" xml:space="preserve">‖ Reperiatur in recta AZ (quæ ſanè curvæ <lb/>diameter eſt) punctum aliquod, ut P, à quo ſi ordinatim applicetur <lb/>PM, &amp; </s>
  <s xml:id="echoid-s9215" xml:space="preserve">ducatur tangens MT, rectæ AZ occurrens in T, ſit in-<lb/>tercepta TP æqualis ipſi PM; </s>
  <s xml:id="echoid-s9216" xml:space="preserve">tum ſumatur in ZA protractâ recta <lb/>AS = AP. </s>
  <s xml:id="echoid-s9217" xml:space="preserve">Dico factum.</s>
  <s xml:id="echoid-s9218" xml:space="preserve"/>
</p>
<div xml:id="echoid-div233" type="float" level="2" n="12">
<note position="right" xlink:label="note-0213-02" xlink:href="note-0213-02a" xml:space="preserve">Fig. 22.</note>
</div>
<p>
  <s xml:id="echoid-s9219" xml:space="preserve">Nam quoniam SA = AP, concipiet mobile deſcendens ab S in <lb/>A tantum impetum, quantum ab A ad P curvam deſcribendo (ponitur <lb/>enim increſcentis velocitatis motus utrobique prorſus idem) iſte verò <lb/>impetus æquatur impetui, quo mobile à T deſcendens uniformi motu <lb/>percurret rectam TP, eodem tempore quo recta AZ uniformiter
<pb o="36" file="0214" n="229" rhead=""/>
lata, pèrque motum iſtum in curva deſcribenda conſpirans, percurrit <lb/>rectam PM. </s>
  <s xml:id="echoid-s9220" xml:space="preserve">Cùm igitur ſint TP, PM ex conſtructione pares, <lb/>adeóque velocitates motuum, quibus ſimul peraguntur, æquales; <lb/></s>
  <s xml:id="echoid-s9221" xml:space="preserve">etiam motus deſcenſivus in P, vel M æquabitur motui tranſverſo, cur-<lb/>vam deſcribenti, hoc eſt motûs ab S ad A velocitas in A eidemæquatur. </s>
  <s xml:id="echoid-s9222" xml:space="preserve"><lb/>Ergo punctum S eſt id ipſum, quod inveniri debuit, &amp; </s>
  <s xml:id="echoid-s9223" xml:space="preserve">abſolutum eſt <lb/>
<anchor type="note" xlink:label="note-0214-01a" xlink:href="note-0214-01"/>
propoſitum.</s>
  <s xml:id="echoid-s9224" xml:space="preserve">| Exemplo ſit _parabola_, quæ facta concipitur ex motu <lb/>uniformi horizontali, &amp; </s>
  <s xml:id="echoid-s9225" xml:space="preserve">deſcenſivo pariter accelerato; </s>
  <s xml:id="echoid-s9226" xml:space="preserve">tum punctum <lb/>P ità facilè per _Analyſin_ inveſtigatur. </s>
  <s xml:id="echoid-s9227" xml:space="preserve">Sit recta R _datæ parabo<unsure/>læ_ <lb/>_rectuns<unsure/> latus._ </s>
  <s xml:id="echoid-s9228" xml:space="preserve">Eſt igitur ex _parabolæ_ natura, R x AP. </s>
  <s xml:id="echoid-s9229" xml:space="preserve">= PMq <lb/>= TPq (exhypotheſi modi noſtri generalis.) </s>
  <s xml:id="echoid-s9230" xml:space="preserve">Item, ex parabolæ <lb/>nota proprietate eſt TPq = 4 APq. </s>
  <s xml:id="echoid-s9231" xml:space="preserve">Ergo eſt R x AP = 4 APq. <lb/></s>
  <s xml:id="echoid-s9232" xml:space="preserve">Adeóque R = 4AP; </s>
  <s xml:id="echoid-s9233" xml:space="preserve">vel {1/4} R = AP = SA. </s>
  <s xml:id="echoid-s9234" xml:space="preserve">Nimirum ita _Gali-_ <lb/>_læus_ determinavit. </s>
  <s xml:id="echoid-s9235" xml:space="preserve">In hoc autem caſu puncta T, S coincidunt. </s>
  <s xml:id="echoid-s9236" xml:space="preserve">Quòd <lb/>ſi rurſus gravia juxta _triplicatam temporum rationem_ velocitate creſcen-<lb/>do deſcendant, adeóque motus ipſorum talis cum uniformi tranſverſo <lb/>compoſitus _parabolam cubicam_ deſcribat, &amp; </s>
  <s xml:id="echoid-s9237" xml:space="preserve">ſit R iſtius curvæ _para-_ <lb/>_meter_, erit eo in caſù SA = √ {R q/27} nam ex hujuſce curvæ proprie-<lb/>tate eſt R q AP = PM cub. </s>
  <s xml:id="echoid-s9238" xml:space="preserve">Et ex hujus regulæ generalis præſcripto <lb/>eſt PM = TP, adeóque PM cub. </s>
  <s xml:id="echoid-s9239" xml:space="preserve">= TP cub. </s>
  <s xml:id="echoid-s9240" xml:space="preserve">Denique quoniam <lb/>in hujuſmodi _parabola_ tangentis intercepta ſemper triſecatur à vertice <lb/>(nimirum ut ſit AP = {1/3} TP) eſt TP cub. </s>
  <s xml:id="echoid-s9241" xml:space="preserve">= 27 AP cub. </s>
  <s xml:id="echoid-s9242" xml:space="preserve">Erit <lb/>igitur R q AP = 27 AP cub. </s>
  <s xml:id="echoid-s9243" xml:space="preserve">Adeóque R q = 27 APq; </s>
  <s xml:id="echoid-s9244" xml:space="preserve">vel <lb/>{Rq/27} = APq = SAq. </s>
  <s xml:id="echoid-s9245" xml:space="preserve">In reliquis ſimili ratione procedentes <lb/>aſſequemur propoſitum. </s>
  <s xml:id="echoid-s9246" xml:space="preserve">Poſſent opinor &amp; </s>
  <s xml:id="echoid-s9247" xml:space="preserve">hinc nedum pleræque <lb/>_Galilæipoſitiones_ huic affines, &amp; </s>
  <s xml:id="echoid-s9248" xml:space="preserve">hanc attingentes materiam utcun-<lb/>
<anchor type="handwritten" xlink:label="hd-0214-01a" xlink:href="hd-0214-01"/>
que deduci, ſed &amp; </s>
  <s xml:id="echoid-s9249" xml:space="preserve">generaliores reddi, vel ad alia curvas omnigenas <lb/>extendi. </s>
  <s xml:id="echoid-s9250" xml:space="preserve">Verùm parco pluribus, hoc _ſpecimine_ (quoad iſta) con-<lb/>tentus; </s>
  <s xml:id="echoid-s9251" xml:space="preserve">huc non niſi per tranſcurſum adducto. </s>
  <s xml:id="echoid-s9252" xml:space="preserve">Ad alia pergo præ-<lb/>dictis cohærentia.</s>
  <s xml:id="echoid-s9253" xml:space="preserve"/>
</p>
<div xml:id="echoid-div234" type="float" level="2" n="13">
<note position="left" xlink:label="note-0214-01" xlink:href="note-0214-01a" xml:space="preserve">Fig. 22.</note>
  <handwritten xlink:label="hd-0214-01" xlink:href="hd-0214-01a"/>
</div>
<p>
  <s xml:id="echoid-s9254" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s9255" xml:space="preserve">Si ad rectam lineam applicetur _planæ ſuperficies_, cujus <lb/>ſingulæ quæque partes applicatis ad iſtam rectam parallelis inter-<lb/>ceptæ proportionales ſint rectis ad rectam AY ſimpliciter diviſam <lb/>applicatis (ad AZ nempe parallelis.) </s>
  <s xml:id="echoid-s9256" xml:space="preserve">Hujuſce ſuperficiei ad paral-<lb/>lelogrammum æquealtum, ſuper eadem baſe conſtitutum, proportio <lb/>proportionem indicabit ipſarum AP; </s>
  <s xml:id="echoid-s9257" xml:space="preserve">TP, à puncto P vertici, tan-<lb/>gentique interjectarum.</s>
  <s xml:id="echoid-s9258" xml:space="preserve"/>
</p>
<pb o="37" file="0215" n="230" rhead=""/>
<p>
  <s xml:id="echoid-s9259" xml:space="preserve">Ut ſi ad rectam α δ applicetur plana ſuperficies α δ μ, &amp; </s>
  <s xml:id="echoid-s9260" xml:space="preserve">utcun-<lb/>
<anchor type="note" xlink:label="note-0215-01a" xlink:href="note-0215-01"/>
que divisâ AD punctis B, C, ſimilitérque dicisâ rectâ α δ punctis <lb/>
<anchor type="note" xlink:label="note-0215-02a" xlink:href="note-0215-02"/>
que divisâ AD punctis B, C, ſimilitérque dicisâ rectâ α δ punctis <lb/>β, γ, fuerit ut BM ad CM ità ſuperficies β α μ, ad ſuperficiem <lb/>γ α μ, &amp; </s>
  <s xml:id="echoid-s9261" xml:space="preserve">hoc in comparationibus univerſis taliter inſtitutis contingat; <lb/></s>
  <s xml:id="echoid-s9262" xml:space="preserve">_completo parallelogrammo α δ μ φ, ſe habebit recta_ AP _adrectam_ TP <lb/>_ut ſuperficies αδ μ adl<unsure/> parallelogrammum_ α δ μ φ. </s>
  <s xml:id="echoid-s9263" xml:space="preserve">Et enim ſi recta <lb/>α δ commune tempus defignare concipiatur, quo recta AD motu <lb/>æquabili, rectáque DM motu continuè accelerato tranſiguntur, <lb/>recta δ μ bene deſignabit velocitatem hujus definiti temporis maxi-<lb/>mam, quam habet punctum deſcendens in curvæ puncto M infimo; </s>
  <s xml:id="echoid-s9264" xml:space="preserve"><lb/>hoc eſt velocitatem quâ recta TP uniformiter decurritur eodem tem-<lb/>pore; </s>
  <s xml:id="echoid-s9265" xml:space="preserve">quapropter (ut antehac commonſtratum eſt.) </s>
  <s xml:id="echoid-s9266" xml:space="preserve">_Parallelogram-_ <lb/>_mum_ α δ μ φ optimè _Spatium_ repræſentabit, quod hâc eâdem per-<lb/>manente velocitate per totum tempus α δ uniformiter deſcribitur, <lb/>hoc eſt ipſam rectam TP. </s>
  <s xml:id="echoid-s9267" xml:space="preserve">Cu<unsure/>m igitur, ex hypotheſis præſtratæ con <lb/>ditione, figura δ α μ rectam DM, vel AP, repræſentet, erit ut figura <lb/>δ αμ ad parallelogrammum α δ μ φ, ità AP ad TP; </s>
  <s xml:id="echoid-s9268" xml:space="preserve">cognitáque <lb/>proinde modo quovis iſtâ proportione, ſimul hæc innoteſcet; </s>
  <s xml:id="echoid-s9269" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9270" xml:space="preserve">re-<lb/>ciprocè. </s>
  <s xml:id="echoid-s9271" xml:space="preserve">Exemplo res manifeſtior evadet uno, vel altero. </s>
  <s xml:id="echoid-s9272" xml:space="preserve">Propoſita <lb/>curva ſit _parabola quadratica_, ſeu in qua rectæ BM, CM ſe <lb/>habent, ut quadrata ex AB, AC, hoc eſt ut quadrata ex α β, α γ. </s>
  <s xml:id="echoid-s9273" xml:space="preserve"><lb/>Ergò ſi figura α δ μ ſit triangulum, id optimè quadrabit huic negotio. </s>
  <s xml:id="echoid-s9274" xml:space="preserve"><lb/>Nam eo ſuppoſito ſemper triangula βαμ, γαμ proportionalia erunt <lb/>quadratis ex α β, αγ, hoc eſt rectis BM; </s>
  <s xml:id="echoid-s9275" xml:space="preserve">CM. </s>
  <s xml:id="echoid-s9276" xml:space="preserve">Quoniam verò <lb/>triangulum δ α μ parallelogrammi δ α φ μ eſt ſubduplum, erit <lb/>recta AP quoque rectæ TP ſubdupla; </s>
  <s xml:id="echoid-s9277" xml:space="preserve">quod ità ſe habere demon-<lb/>ſtratum habetur in _conicis elementis_, &amp; </s>
  <s xml:id="echoid-s9278" xml:space="preserve">paſſim agnoſcitur. </s>
  <s xml:id="echoid-s9279" xml:space="preserve">Sit rurſus <lb/>curva AMM _parabola cubica_; </s>
  <s xml:id="echoid-s9280" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9281" xml:space="preserve">quoniam in ea rectæ BM, CM <lb/>ſe habent ut cubi rectarum AB, AC, hoc eſt ut cubi rectarum α β, <lb/>α γ; </s>
  <s xml:id="echoid-s9282" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9283" xml:space="preserve">ſi _ſuperficies α δ μ fuerit complementum ſemiparabolicæ qua-_ <lb/>_draticæ portionis, trilinea α β μ, α γ μ cubis ex α β, α γ proportionalia_ <lb/>_erunt_ (ut à _Pappo_, ac aliis oſtenditur, &amp; </s>
  <s xml:id="echoid-s9284" xml:space="preserve">ex _Archimidea parabolæ_ <lb/>_dimenſione_ quàm facillimè deducitur) itaque negotio propoſito quàm <lb/>rectiſſimè adaptetur _parabola quadratica_; </s>
  <s xml:id="echoid-s9285" xml:space="preserve">cúmque conſtiterit ali-<lb/>undè tum figuram α δ μ ſubtriplam fore parallelogrammi α δ μ φ; </s>
  <s xml:id="echoid-s9286" xml:space="preserve"><lb/>erit etiam juxta regulæ jam aſſignatæ præſcriptum recta AP quoque <lb/>ſubtripla rectæ TP. </s>
  <s xml:id="echoid-s9287" xml:space="preserve">De qua concluſione ſatis convenit inter _Geo-_ <lb/>_metras_.</s>
  <s xml:id="echoid-s9288" xml:space="preserve"/>
</p>
<div xml:id="echoid-div235" type="float" level="2" n="14">
<note position="right" xlink:label="note-0215-01" xlink:href="note-0215-01a" xml:space="preserve">Fig. 23, 24.</note>
<note style="it" position="right" xlink:label="note-0215-02" xlink:href="note-0215-02a" xml:space="preserve">Hæc poſthac <lb/>γτωμετριηίο <lb/>τερον demo@-<lb/>ſtrata haben-<lb/>tur.</note>
</div>
<pb o="38" file="0216" n="231" rhead=""/>
<p>
  <s xml:id="echoid-s9289" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s9290" xml:space="preserve">Huic ſuppar modus dictas @rectas AP, TP comparandi <lb/>tali _Theoremate_ continetur: </s>
  <s xml:id="echoid-s9291" xml:space="preserve">Si ad rectam aliquam lineam (hoc eſt <lb/>ad cjus ſingula quæque puncta) applicentur rectæ lineæ parallelæ, ad <lb/>
<anchor type="note" xlink:label="note-0216-01a" xlink:href="note-0216-01"/>
rectam AD conſimiliter diviſam applicatarum differentiis proportio-<lb/>nales, reſultantis hinc plani ad parallelogrammum æque altum, ad <lb/>eandémque baſin poſitum, rectarum AP, TP proportionem exhi-<lb/>bebit. </s>
  <s xml:id="echoid-s9292" xml:space="preserve">Ut ſi rectæ AD, α δ ſimiliter (in partes ſcilicet æquales in-<lb/>definitè multas) dividantur; </s>
  <s xml:id="echoid-s9293" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9294" xml:space="preserve">rectæ β μ, γ μ, δ μ rectis BM, NM, <lb/>OM (quæ differentiæ ſunt rectarum ad AD applicatarum, incipi-<lb/>endo à puncto A) proportionales ſint, erit ut figura α δ μ ad paral-<lb/>lelogrammum α<unsure/> δ μ φ, ita AP ad TP. </s>
  <s xml:id="echoid-s9295" xml:space="preserve">Cum enim recta quæpiam <lb/>ex applicatis ad AD; </s>
  <s xml:id="echoid-s9296" xml:space="preserve">puta _v.</s>
  <s xml:id="echoid-s9297" xml:space="preserve">g._ </s>
  <s xml:id="echoid-s9298" xml:space="preserve">DM æquetur omnibus ſeipsâ mi-<lb/>norum differentiis (ipſis nempe BM, NM, OM) &amp; </s>
  <s xml:id="echoid-s9299" xml:space="preserve">trilineum <lb/>α δ μ conſtituatur è rectis β μ, γ μ δ μ eâdem proportione creſcen-<lb/>tibus; </s>
  <s xml:id="echoid-s9300" xml:space="preserve">ut &amp; </s>
  <s xml:id="echoid-s9301" xml:space="preserve">recta CM æquatur ipſis BM, NM; </s>
  <s xml:id="echoid-s9302" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9303" xml:space="preserve">ei reſpondens <lb/>trilineum α γ μ quaſi conflatur è parallelis β μ, γ μ pari ratione <lb/>creſcentibus; </s>
  <s xml:id="echoid-s9304" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9305" xml:space="preserve">hoc ſemper eveniat; </s>
  <s xml:id="echoid-s9306" xml:space="preserve">omnino patet trilinea α δ μ, <lb/>α γ μ, α β μ rectis DM, CM, BM proportionari; </s>
  <s xml:id="echoid-s9307" xml:space="preserve">proindéque <lb/>modum hunc in priorem recidere; </s>
  <s xml:id="echoid-s9308" xml:space="preserve">nec ab eo reipsâ differre. </s>
  <s xml:id="echoid-s9309" xml:space="preserve">Notetur <lb/>autem hic rectas β μ, γ μ, δ μ velocitates repræſentare, quas pun-<lb/>ctum mobile curvam delineans obtinet in reſpectivis ejus punctis M; <lb/></s>
  <s xml:id="echoid-s9310" xml:space="preserve">ut &amp; </s>
  <s xml:id="echoid-s9311" xml:space="preserve">trilinea α β μ, α γ μ, α δ μ velocitates aggregatas exhibent ab <lb/>initio ad definita reſpectiva temporis inſtantia; </s>
  <s xml:id="echoid-s9312" xml:space="preserve">quibus (ut jam olim <lb/>præmonitum) reſpondentia ſpatia BM, CM, DM proportionantur.</s>
  <s xml:id="echoid-s9313" xml:space="preserve"/>
</p>
<div xml:id="echoid-div236" type="float" level="2" n="15">
<note position="left" xlink:label="note-0216-01" xlink:href="note-0216-01a" xml:space="preserve">Fig. 23, 24.</note>
</div>
<p>
  <s xml:id="echoid-s9314" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s9315" xml:space="preserve">E ſupradictis porrò conſectatur, quòd ſi _Circulus,_ <lb/>_Ellipſis_, ejuſmodíque curvæ recurrentes hoc progenitæ concipiantur <lb/>modo, punctum eas deſcribens infinitam in recursûs puncto veloci-<lb/>tatem habebit. </s>
  <s xml:id="echoid-s9316" xml:space="preserve">Nempe ſi quadrans AFM ità generetur; </s>
  <s xml:id="echoid-s9317" xml:space="preserve">quoniam <lb/>
<anchor type="note" xlink:label="note-0216-02a" xlink:href="note-0216-02"/>
tangens TM diametro AZ eſt parallela, nec illa proinde cum hac <lb/>niſi ad infinitam diſtantiam convenit; </s>
  <s xml:id="echoid-s9318" xml:space="preserve">ergô velocitas in M ad veloci-<lb/>tatem uniformis motûs per AY ſe habebit, ut infinita recta ad ipſam <lb/>PM; </s>
  <s xml:id="echoid-s9319" xml:space="preserve">unde velocitas iſta ad M prorſus infinita ſit oportet. </s>
  <s xml:id="echoid-s9320" xml:space="preserve">Ità quidem <lb/>
<anchor type="note" xlink:label="note-0216-03a" xlink:href="note-0216-03"/>
quoad hujuſmodi curvas; </s>
  <s xml:id="echoid-s9321" xml:space="preserve">at quoad alias ad infinitum ſenſim continu-<lb/>atas (quales _parabolæ &amp; </s>
  <s xml:id="echoid-s9322" xml:space="preserve">byperbolæ_) deſcendentis puncti velocitas in <lb/>quovis deſignato curvæ puncto finita eſt. </s>
  <s xml:id="echoid-s9323" xml:space="preserve">Verùm his omiſſis ad alias <lb/>propoſitæ curvæ proprietates exponendas progrediamur.</s>
  <s xml:id="echoid-s9324" xml:space="preserve"/>
</p>
<div xml:id="echoid-div237" type="float" level="2" n="16">
<note position="left" xlink:label="note-0216-02" xlink:href="note-0216-02a" xml:space="preserve">Fig. 25.</note>
<note position="left" xlink:label="note-0216-03" xlink:href="note-0216-03a" xml:space="preserve">II. _preced_.</note>
</div>
<pb o="39" file="0217" n="232" rhead=""/>
<p>
  <s xml:id="echoid-s9325" xml:space="preserve">IN deducendis è propoſitâ generatione curvarum affectionibus etiam-<lb/>num progredimur.</s>
  <s xml:id="echoid-s9326" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s9327" xml:space="preserve">_I._ </s>
  <s xml:id="echoid-s9328" xml:space="preserve">Anguli, qui fiunt ab applicatis &amp; </s>
  <s xml:id="echoid-s9329" xml:space="preserve">tangentibus ad diverſa curvæ <lb/>puncta, ſibimet inæquales ſunt; </s>
  <s xml:id="echoid-s9330" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9331" xml:space="preserve">minores ſunt illi qui puncto A (ſe@@ <lb/>vertici) propiores ſnnt.</s>
  <s xml:id="echoid-s9332" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9333" xml:space="preserve">Tangant rectæ TM, XN; </s>
  <s xml:id="echoid-s9334" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9335" xml:space="preserve">ad AY parallelæ ſint MP, NQ, <lb/>dico fore angulum PMT minorem angulo QNX.</s>
  <s xml:id="echoid-s9336" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9337" xml:space="preserve">Nam producta recta TM occurret ipſi QN extra curvam pro-<lb/>
<anchor type="note" xlink:label="note-0217-01a" xlink:href="note-0217-01"/>
tractæ, puta ad E. </s>
  <s xml:id="echoid-s9338" xml:space="preserve">Item ipſa XN ſecabit applicatam PM extra <lb/>curvam, puta ad H. </s>
  <s xml:id="echoid-s9339" xml:space="preserve">Manifeſtum eſt autem cùm puncta H, N ſint <lb/>ad alias, ac alias partes rectæ ME, rectas ME, NH ſeſe inter-<lb/>ſecare inter parallelas PH, QE; </s>
  <s xml:id="echoid-s9340" xml:space="preserve">quare major eſt angulus externus <lb/>QNX interno QET, hoc eſt angulo PMT: </s>
  <s xml:id="echoid-s9341" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s9342" xml:space="preserve">E. </s>
  <s xml:id="echoid-s9343" xml:space="preserve">D.</s>
  <s xml:id="echoid-s9344" xml:space="preserve"/>
</p>
<div xml:id="echoid-div238" type="float" level="2" n="17">
<note position="right" xlink:label="note-0217-01" xlink:href="note-0217-01a" xml:space="preserve">Fig. 26.</note>
</div>
<p style="it">
  <s xml:id="echoid-s9345" xml:space="preserve">_II. </s>
  <s xml:id="echoid-s9346" xml:space="preserve">Hinc_ poriſmatis _loco habetur_ tangentes ſe interſecare inter or-<lb/>dinatim applicatas per puncta contactuum; </s>
  <s xml:id="echoid-s9347" xml:space="preserve">velut ad F, inter PM, <lb/>QN protenſas.</s>
  <s xml:id="echoid-s9348" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9349" xml:space="preserve">III. </s>
  <s xml:id="echoid-s9350" xml:space="preserve">Item _angulum PTM angulo QXN majorem eſſe_; </s>
  <s xml:id="echoid-s9351" xml:space="preserve">(ex <lb/>ternum ſcilicet interno.)</s>
  <s xml:id="echoid-s9352" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9353" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s9354" xml:space="preserve">Item patet vertici propiores applicatas (proindéque rectas <lb/>quaſvis aliis parallelas) cur<unsure/>væ obliquiùs incidere quàm remotiores.</s>
  <s xml:id="echoid-s9355" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9356" xml:space="preserve">Cæterùm iſta jam olim de _sectionibus Conicis_ oſtenderat _Apollonius_, <lb/>ut in edito nuper VI _conicorv<unsure/>m libro_ eſt videre</s>
</p>
<p>
  <s xml:id="echoid-s9357" xml:space="preserve">V. </s>
  <s xml:id="echoid-s9358" xml:space="preserve">In figura præcedente (poſito applicationis angulum TAY <lb/>rectum eſſe, vel obtuſum) _dico curvæ arcum MN rectâ NH, ma-_ <lb/>_jorem eſſe; </s>
  <s xml:id="echoid-s9359" xml:space="preserve">rectâverò ME minorem_.</s>
  <s xml:id="echoid-s9360" xml:space="preserve"/>
</p>
<pb o="40" file="0218" n="233" rhead=""/>
<p>
  <s xml:id="echoid-s9361" xml:space="preserve">Nam connectatur ſubtenſa MN, ducatúrque recta NR ad ZA <lb/>parallela. </s>
  <s xml:id="echoid-s9362" xml:space="preserve">Et quoniam angulus XPH non minor eſt recto, erit, <lb/>eo major externus, NHP obtuſus. </s>
  <s xml:id="echoid-s9363" xml:space="preserve">Ergò recta NM major eſt quàm <lb/>NH. </s>
  <s xml:id="echoid-s9364" xml:space="preserve">Itaque magis arcus, arcus NH major eſt quàm ipsâ NH: <lb/></s>
  <s xml:id="echoid-s9365" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s9366" xml:space="preserve">E. </s>
  <s xml:id="echoid-s9367" xml:space="preserve">D.</s>
  <s xml:id="echoid-s9368" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9369" xml:space="preserve">Item, quoniam ang. </s>
  <s xml:id="echoid-s9370" xml:space="preserve">RNE ipſi XQE par haud minor eſt recto, <lb/>erit RE &amp;</s>
  <s xml:id="echoid-s9371" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9372" xml:space="preserve">RN. </s>
  <s xml:id="echoid-s9373" xml:space="preserve">quare MR + RE &amp;</s>
  <s xml:id="echoid-s9374" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9375" xml:space="preserve">MR + RN. </s>
  <s xml:id="echoid-s9376" xml:space="preserve">hoc eſt <lb/>ME &amp;</s>
  <s xml:id="echoid-s9377" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9378" xml:space="preserve">MR + RN. </s>
  <s xml:id="echoid-s9379" xml:space="preserve">Eſt autem (ex _Arcbimedæis_ aſſumptis) MR <lb/>+ RN &amp;</s>
  <s xml:id="echoid-s9380" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9381" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s9382" xml:space="preserve">MN. </s>
  <s xml:id="echoid-s9383" xml:space="preserve">ergò magìs eſt ME &amp;</s>
  <s xml:id="echoid-s9384" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9385" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s9386" xml:space="preserve">MN ∴</s>
</p>
<p>
  <s xml:id="echoid-s9387" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s9388" xml:space="preserve">Perutilis eſt hæc propoſitio in _tangentium demonſtra@ionibus_ <lb/>_expediendis_. </s>
  <s xml:id="echoid-s9389" xml:space="preserve">Etenim hinc couſectatur, ſi arcus MN indefinitè par-<lb/>vus ponatur, ejuſce loco alterutram tangentis particulam ME, vel <lb/>NH tutò ſubſtitui.</s>
  <s xml:id="echoid-s9390" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9391" xml:space="preserve">_Speciminis_ hîc loco _metbodum proponam generalem cycloidum om-_ <lb/>_nium, &amp; </s>
  <s xml:id="echoid-s9392" xml:space="preserve">conſimili modo deſcriptarum curvarum tangentes determi-_ <lb/>_nandi_, hinc petitâ demonſtratione munitam.</s>
  <s xml:id="echoid-s9393" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9394" xml:space="preserve">Recta AY ſibi parallelè deportata quamcunque curvam ad eaſdem <lb/>
<anchor type="note" xlink:label="note-0218-01a" xlink:href="note-0218-01"/>
partes convexam aut cavam, APX perambulet uniformi motu (ſci-<lb/>licet ut æquales curvæ partes æqualibus tranſigat temporibus) eodém-<lb/>que ſimul tempore punctum aliquod ab A per AY etiam uniformiter <lb/>feratur; </s>
  <s xml:id="echoid-s9395" xml:space="preserve">ab hoc puncto taliter moto progignetur curva AMZ; <lb/></s>
  <s xml:id="echoid-s9396" xml:space="preserve">cujus ad datum quodcunque punctum M tangentem oportet determi-<lb/>nare. </s>
  <s xml:id="echoid-s9397" xml:space="preserve">Ut hoc fiat, ducatur recta MP ad AY parallela, curvam <lb/>APX ſecans in P; </s>
  <s xml:id="echoid-s9398" xml:space="preserve">pérque P ducatur recta PE curvam APX con-<lb/>tingens; </s>
  <s xml:id="echoid-s9399" xml:space="preserve">huic verò per M ducatur parallela MH; </s>
  <s xml:id="echoid-s9400" xml:space="preserve">ínque hac ſumatur <lb/>punctum quodpiam R, &amp; </s>
  <s xml:id="echoid-s9401" xml:space="preserve">ducatur RS ad PM parallela; </s>
  <s xml:id="echoid-s9402" xml:space="preserve">tum fiat <lb/>ut curva AP ad rectam PM (hoc eſt ut unus motus uniformis ad <lb/>alterum) ità MR ad RS; </s>
  <s xml:id="echoid-s9403" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9404" xml:space="preserve">connectatur MS. </s>
  <s xml:id="echoid-s9405" xml:space="preserve">hæc curvam AMZ <lb/>continget. </s>
  <s xml:id="echoid-s9406" xml:space="preserve">Sumatur enim in hac curva punctum quodvis Z, per quod <lb/>ducatur recta ZX ad MP parallela, ſecans curvam APX in X, <lb/>ejúſque tangentem in E; </s>
  <s xml:id="echoid-s9407" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9408" xml:space="preserve">huic parallelam MR in H; </s>
  <s xml:id="echoid-s9409" xml:space="preserve">ipsámque <lb/>demum MS in K. </s>
  <s xml:id="echoid-s9410" xml:space="preserve">Sit autem primò punctum Z ſupra M verſus A; </s>
  <s xml:id="echoid-s9411" xml:space="preserve"><lb/>unde recta PE &amp;</s>
  <s xml:id="echoid-s9412" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9413" xml:space="preserve">arc PX. </s>
  <s xml:id="echoid-s9414" xml:space="preserve">adeóque PA. </s>
  <s xml:id="echoid-s9415" xml:space="preserve">PE &amp;</s>
  <s xml:id="echoid-s9416" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9417" xml:space="preserve">arc PA. </s>
  <s xml:id="echoid-s9418" xml:space="preserve">PX:</s>
  <s xml:id="echoid-s9419" xml:space="preserve">: <lb/>PM. </s>
  <s xml:id="echoid-s9420" xml:space="preserve">PM - XZ:</s>
  <s xml:id="echoid-s9421" xml:space="preserve">: PM. </s>
  <s xml:id="echoid-s9422" xml:space="preserve">EH - XZ:</s>
  <s xml:id="echoid-s9423" xml:space="preserve">: PM. </s>
  <s xml:id="echoid-s9424" xml:space="preserve">ZH - EX &amp;</s>
  <s xml:id="echoid-s9425" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9426" xml:space="preserve"><lb/>PM. </s>
  <s xml:id="echoid-s9427" xml:space="preserve">ZH. </s>
  <s xml:id="echoid-s9428" xml:space="preserve">quare permutatim erit PA. </s>
  <s xml:id="echoid-s9429" xml:space="preserve">PM &amp;</s>
  <s xml:id="echoid-s9430" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9431" xml:space="preserve">PE. </s>
  <s xml:id="echoid-s9432" xml:space="preserve">ZH. </s>
  <s xml:id="echoid-s9433" xml:space="preserve">eſt <lb/>autem PA. </s>
  <s xml:id="echoid-s9434" xml:space="preserve">PM:</s>
  <s xml:id="echoid-s9435" xml:space="preserve">: MR. </s>
  <s xml:id="echoid-s9436" xml:space="preserve">RS:</s>
  <s xml:id="echoid-s9437" xml:space="preserve">: MH. </s>
  <s xml:id="echoid-s9438" xml:space="preserve">KH:</s>
  <s xml:id="echoid-s9439" xml:space="preserve">: PE. </s>
  <s xml:id="echoid-s9440" xml:space="preserve">HK. </s>
  <s xml:id="echoid-s9441" xml:space="preserve">ergò <lb/>PE. </s>
  <s xml:id="echoid-s9442" xml:space="preserve">HK. </s>
  <s xml:id="echoid-s9443" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s9444" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9445" xml:space="preserve">PE. </s>
  <s xml:id="echoid-s9446" xml:space="preserve">ZH quare HK &amp;</s>
  <s xml:id="echoid-s9447" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9448" xml:space="preserve">ZH. </s>
  <s xml:id="echoid-s9449" xml:space="preserve">eſt autem punctum <lb/>H extra curvam AZM; </s>
  <s xml:id="echoid-s9450" xml:space="preserve">ob EZ &amp;</s>
  <s xml:id="echoid-s9451" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9452" xml:space="preserve">XZ &amp;</s>
  <s xml:id="echoid-s9453" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9454" xml:space="preserve">PM = EH. </s>
  <s xml:id="echoid-s9455" xml:space="preserve">ergò <lb/>palàm eſt punctum K extra curvam AZM exiſtere. </s>
  <s xml:id="echoid-s9456" xml:space="preserve">Sit vero ſecundò
<pb o="41" file="0219" n="234" rhead=""/>
punctum Zinfra punctum M; </s>
  <s xml:id="echoid-s9457" xml:space="preserve">erit tum recta PE major arcu PX; <lb/></s>
  <s xml:id="echoid-s9458" xml:space="preserve">
<anchor type="note" xlink:label="note-0219-01a" xlink:href="note-0219-01"/>
unde arc PA. </s>
  <s xml:id="echoid-s9459" xml:space="preserve">PE &amp;</s>
  <s xml:id="echoid-s9460" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9461" xml:space="preserve">arc PA. </s>
  <s xml:id="echoid-s9462" xml:space="preserve">PX:</s>
  <s xml:id="echoid-s9463" xml:space="preserve">: PM. </s>
  <s xml:id="echoid-s9464" xml:space="preserve">XZ - PM:</s>
  <s xml:id="echoid-s9465" xml:space="preserve">: <lb/>PM. </s>
  <s xml:id="echoid-s9466" xml:space="preserve">XZ - EH:</s>
  <s xml:id="echoid-s9467" xml:space="preserve">: PM. </s>
  <s xml:id="echoid-s9468" xml:space="preserve">XE + XZ &amp;</s>
  <s xml:id="echoid-s9469" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9470" xml:space="preserve">PM. </s>
  <s xml:id="echoid-s9471" xml:space="preserve">HZ. </s>
  <s xml:id="echoid-s9472" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9473" xml:space="preserve">viciſſim <lb/>PA. </s>
  <s xml:id="echoid-s9474" xml:space="preserve">PM &amp;</s>
  <s xml:id="echoid-s9475" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9476" xml:space="preserve">PE. </s>
  <s xml:id="echoid-s9477" xml:space="preserve">HZ. </s>
  <s xml:id="echoid-s9478" xml:space="preserve">Verum ut priùs) eſt PA. </s>
  <s xml:id="echoid-s9479" xml:space="preserve">PM:</s>
  <s xml:id="echoid-s9480" xml:space="preserve">: PE. </s>
  <s xml:id="echoid-s9481" xml:space="preserve">HK. <lb/></s>
  <s xml:id="echoid-s9482" xml:space="preserve">ergò PE. </s>
  <s xml:id="echoid-s9483" xml:space="preserve">HK &amp;</s>
  <s xml:id="echoid-s9484" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9485" xml:space="preserve">PE. </s>
  <s xml:id="echoid-s9486" xml:space="preserve">HZ; </s>
  <s xml:id="echoid-s9487" xml:space="preserve">proptereáque HK &amp;</s>
  <s xml:id="echoid-s9488" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9489" xml:space="preserve">HZ; </s>
  <s xml:id="echoid-s9490" xml:space="preserve">ro<unsure/>rſus <lb/>itaque liquet Punctum K extra curvam exiſtere. </s>
  <s xml:id="echoid-s9491" xml:space="preserve">Tota proinde recta <lb/>MKZ extra curvam verſatur; </s>
  <s xml:id="echoid-s9492" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9493" xml:space="preserve">eam tangit ad M: </s>
  <s xml:id="echoid-s9494" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s9495" xml:space="preserve">E. </s>
  <s xml:id="echoid-s9496" xml:space="preserve">D. </s>
  <s xml:id="echoid-s9497" xml:space="preserve"><lb/>In tranſcurſu hoc. </s>
  <s xml:id="echoid-s9498" xml:space="preserve">ad alias curvæ noſtræ paſſiones revertamur.</s>
  <s xml:id="echoid-s9499" xml:space="preserve"/>
</p>
<div xml:id="echoid-div239" type="float" level="2" n="18">
<note position="left" xlink:label="note-0218-01" xlink:href="note-0218-01a" xml:space="preserve">Fig. 27.</note>
<note position="right" xlink:label="note-0219-01" xlink:href="note-0219-01a" xml:space="preserve">Fig. 27.</note>
</div>
<p>
  <s xml:id="echoid-s9500" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s9501" xml:space="preserve">Si tangenti cuipiam (ut ipſi MT) parallela ducatur quæ-<lb/>piam EF (à puncto nempe quopiam E in recta infra punctum T ſum-<lb/>pto) hæc curvæ occurret.</s>
  <s xml:id="echoid-s9502" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9503" xml:space="preserve">Si enim infra punctum M in curva ſumatur punctum quodlibet, &amp; </s>
  <s xml:id="echoid-s9504" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0219-02a" xlink:href="note-0219-02"/>
ab eo duci concipiatur curvam tangens recta; </s>
  <s xml:id="echoid-s9505" xml:space="preserve">huic occurret tangens <lb/>TM infra ordinatam PM. </s>
  <s xml:id="echoid-s9506" xml:space="preserve">ergò recta EF eidem occurret; </s>
  <s xml:id="echoid-s9507" xml:space="preserve">at curvam <lb/>priùs tranſiliat oportet. </s>
  <s xml:id="echoid-s9508" xml:space="preserve">ergò liquet Propoſitum.</s>
  <s xml:id="echoid-s9509" xml:space="preserve"><unsure/></s>
</p>
<div xml:id="echoid-div240" type="float" level="2" n="19">
<note position="right" xlink:label="note-0219-02" xlink:href="note-0219-02a" xml:space="preserve">Fig. 28.</note>
</div>
<p>
  <s xml:id="echoid-s9510" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s9511" xml:space="preserve">Eâdem operâ patet, ſi punctum aſſumptum E puncto T, <lb/>&amp; </s>
  <s xml:id="echoid-s9512" xml:space="preserve">vertici A interjiciatur, rectum EF curvæ bis occurſuram, tam ſupra <lb/>quàm infra contactum M.</s>
  <s xml:id="echoid-s9513" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">con. 1. 27, 28.</note>
<p>
  <s xml:id="echoid-s9514" xml:space="preserve">Operosè conniſus eſt _Apollonius_ hæc de _Sectionibus Conicis_ oſtendere.</s>
  <s xml:id="echoid-s9515" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9516" xml:space="preserve">Cæterùm ad penitus determinandos occurſuum locos _Specialis mo-_ <lb/>_dus ſeu ratio motuum deſcendentis atq; </s>
  <s xml:id="echoid-s9517" xml:space="preserve">tranſverſi cognoſci debet_; </s>
  <s xml:id="echoid-s9518" xml:space="preserve">tunc <lb/>eos _Analyſis_ ſtatim prodet.</s>
  <s xml:id="echoid-s9519" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9520" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s9521" xml:space="preserve">Si duæ rectæ quævis (HM, KN) ad curvam propoſi-<lb/>
<anchor type="note" xlink:label="note-0219-04a" xlink:href="note-0219-04"/>
tam æqualiter inclinentur (hoc eſt æquales cum ejus ad occurſus tan-<lb/>gentibus (puta cum ipſis MT, NX) angulos efficiant) hæ extrorſum <lb/>divergent, ſeu ad partes EF productæ concurrent.</s>
  <s xml:id="echoid-s9522" xml:space="preserve"/>
</p>
<div xml:id="echoid-div241" type="float" level="2" n="20">
<note position="right" xlink:label="note-0219-04" xlink:href="note-0219-04a" xml:space="preserve">Fig. 29.</note>
</div>
<p>
  <s xml:id="echoid-s9523" xml:space="preserve">Nam ducatur ſubtenſa NM; </s>
  <s xml:id="echoid-s9524" xml:space="preserve">hæc utiq; </s>
  <s xml:id="echoid-s9525" xml:space="preserve">ſecundum antedicta cum <lb/>ipſa AZ conveniet, puta ad O. </s>
  <s xml:id="echoid-s9526" xml:space="preserve">Eſt ergò ang OMH &amp;</s>
  <s xml:id="echoid-s9527" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9528" xml:space="preserve">(ang. <lb/></s>
  <s xml:id="echoid-s9529" xml:space="preserve">TMH = ang. </s>
  <s xml:id="echoid-s9530" xml:space="preserve">XNK &amp;</s>
  <s xml:id="echoid-s9531" xml:space="preserve">lt;) </s>
  <s xml:id="echoid-s9532" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s9533" xml:space="preserve">ONK. </s>
  <s xml:id="echoid-s9534" xml:space="preserve">ergo ang. </s>
  <s xml:id="echoid-s9535" xml:space="preserve">HMN + <lb/>MNK &amp;</s>
  <s xml:id="echoid-s9536" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9537" xml:space="preserve">2 rect. </s>
  <s xml:id="echoid-s9538" xml:space="preserve">ergò rectæ HM, KN concurrunt ad partes EF. </s>
  <s xml:id="echoid-s9539" xml:space="preserve"><lb/>Limitandum eſt hoc, intelligendo pares angulos HMA, KNA ad <lb/>eaſdem partes veſari; </s>
  <s xml:id="echoid-s9540" xml:space="preserve">ſeu alterum alteri fore externum interno. </s>
  <s xml:id="echoid-s9541" xml:space="preserve">alias <lb/>cotnrà eveniet.</s>
  <s xml:id="echoid-s9542" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9543" xml:space="preserve">X. </s>
  <s xml:id="echoid-s9544" xml:space="preserve">Si fuerit recta HM _curvæ_ perpendicularis (hoc ejus tan-<lb/>
<anchor type="note" xlink:label="note-0219-05a" xlink:href="note-0219-05"/>
genti MT) &amp; </s>
  <s xml:id="echoid-s9545" xml:space="preserve">in hac ſumatnr quæpiam definita HM; </s>
  <s xml:id="echoid-s9546" xml:space="preserve">erit HM mi-<lb/>
<anchor type="note" xlink:label="note-0219-06a" xlink:href="note-0219-06"/>
nima rectarum omnium, quæ à puncto H duci poſſunt ad curvam.</s>
  <s xml:id="echoid-s9547" xml:space="preserve"/>
</p>
<div xml:id="echoid-div242" type="float" level="2" n="21">
<note position="right" xlink:label="note-0219-05" xlink:href="note-0219-05a" xml:space="preserve">Fig. 30.</note>
<note position="right" xlink:label="note-0219-06" xlink:href="note-0219-06a" xml:space="preserve">_Apoll. V._ 38.&amp;c.</note>
</div>
<pb o="42" file="0220" n="235" rhead=""/>
<p>
  <s xml:id="echoid-s9548" xml:space="preserve">Ducatur enim quævis HO; </s>
  <s xml:id="echoid-s9549" xml:space="preserve">hæc tangenti priùs occurret, puta ad <lb/>R. </s>
  <s xml:id="echoid-s9550" xml:space="preserve">liquet HR majorem eſſe quàm HM; </s>
  <s xml:id="echoid-s9551" xml:space="preserve">multóq; </s>
  <s xml:id="echoid-s9552" xml:space="preserve">magìs eſſe HO <lb/>&amp;</s>
  <s xml:id="echoid-s9553" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9554" xml:space="preserve">HM,</s>
</p>
<p>
  <s xml:id="echoid-s9555" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s9556" xml:space="preserve">Hinc _Circulus Centro_ H per M deſcriptus _curvam_ contin-<lb/>get.</s>
  <s xml:id="echoid-s9557" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9558" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s9559" xml:space="preserve">Etiam inversè, ſi HM minima ſit omnium quæ ab H ad <lb/>curvam duci poſſunt, erit HM curvæ perpendicularis.</s>
  <s xml:id="echoid-s9560" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9561" xml:space="preserve">Nam quoniam HM minima ponitur, circulus centro H, intervallo <lb/>quovis HS, majori quàm HM, curvam ſecabit, &amp; </s>
  <s xml:id="echoid-s9562" xml:space="preserve">proinde tangentem <lb/>MT, hanc puta in R. </s>
  <s xml:id="echoid-s9563" xml:space="preserve">ergò quum ſit HR &amp;</s>
  <s xml:id="echoid-s9564" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9565" xml:space="preserve">HM, non erit angu-<lb/>lus HRM rectus. </s>
  <s xml:id="echoid-s9566" xml:space="preserve">idem de punctis omnibus in recta TM evidens eſt. <lb/></s>
  <s xml:id="echoid-s9567" xml:space="preserve">ergò tangenti perpendicularis non alibi quàm in punctum M cadit.</s>
  <s xml:id="echoid-s9568" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9569" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s9570" xml:space="preserve">Quinetiam ſi recta HM minima ſit omnium quæ ab H <lb/>
<anchor type="note" xlink:label="note-0220-01a" xlink:href="note-0220-01"/>
duci poſſunt, eíq; </s>
  <s xml:id="echoid-s9571" xml:space="preserve">perpendicularis ſit recta TM; </s>
  <s xml:id="echoid-s9572" xml:space="preserve">hæc curvam tan-<lb/>get.</s>
  <s xml:id="echoid-s9573" xml:space="preserve"/>
</p>
<div xml:id="echoid-div243" type="float" level="2" n="22">
<note position="left" xlink:label="note-0220-01" xlink:href="note-0220-01a" xml:space="preserve">Fig. 31.</note>
</div>
<p>
  <s xml:id="echoid-s9574" xml:space="preserve">Nam tangat alia, (ſi fieri poteſt) XM; </s>
  <s xml:id="echoid-s9575" xml:space="preserve">erit igitur XM ad HM <lb/>perpendicularis. </s>
  <s xml:id="echoid-s9576" xml:space="preserve">Unde pares erunt anguli HMX, HMT; </s>
  <s xml:id="echoid-s9577" xml:space="preserve">totum <lb/>&amp; </s>
  <s xml:id="echoid-s9578" xml:space="preserve">pars Q: </s>
  <s xml:id="echoid-s9579" xml:space="preserve">E.</s>
  <s xml:id="echoid-s9580" xml:space="preserve">A.</s>
  <s xml:id="echoid-s9581" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9582" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s9583" xml:space="preserve">Dico porrò minimæ HM propiorem HN remotiore HO <lb/>
<anchor type="note" xlink:label="note-0220-02a" xlink:href="note-0220-02"/>
minorem eſſe.</s>
  <s xml:id="echoid-s9584" xml:space="preserve"/>
</p>
<div xml:id="echoid-div244" type="float" level="2" n="23">
<note position="left" xlink:label="note-0220-02" xlink:href="note-0220-02a" xml:space="preserve">Fig. 32.</note>
</div>
<p>
  <s xml:id="echoid-s9585" xml:space="preserve">Nam ducatur ſubtenſa MN; </s>
  <s xml:id="echoid-s9586" xml:space="preserve">hæc producta curvam tranſgredietur, <lb/>&amp; </s>
  <s xml:id="echoid-s9587" xml:space="preserve">ipſam HO ſecabit, puta in R. </s>
  <s xml:id="echoid-s9588" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9589" xml:space="preserve">quoniam Angulus HMR obtu-<lb/>ſus eſt (major illo nempe, quem tangens cum HM conſtituit ad M) <lb/>erit HNR magìs obtuſus; </s>
  <s xml:id="echoid-s9590" xml:space="preserve">adeôq; </s>
  <s xml:id="echoid-s9591" xml:space="preserve">recta HR &amp;</s>
  <s xml:id="echoid-s9592" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9593" xml:space="preserve">HN &amp; </s>
  <s xml:id="echoid-s9594" xml:space="preserve">magis HO &amp;</s>
  <s xml:id="echoid-s9595" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9596" xml:space="preserve">HN.</s>
  <s xml:id="echoid-s9597" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9598" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s9599" xml:space="preserve">Hinc perſpicuum eſt Circulum quemvis Centro H deſcri-<lb/>ptum, uno tantùm ad eaſdem puncti M partes puncto curvæ occurrere; <lb/></s>
  <s xml:id="echoid-s9600" xml:space="preserve">nec omnino pluries igitur, quàm in duobus punctis.</s>
  <s xml:id="echoid-s9601" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9602" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s9603" xml:space="preserve">Perpendiculari HM parallelæ, ſint rectæ IN, KO; </s>
  <s xml:id="echoid-s9604" xml:space="preserve">ha-<lb/>
<anchor type="note" xlink:label="note-0220-03a" xlink:href="note-0220-03"/>
rum propior IN remotiore KO rectiùs incidet.</s>
  <s xml:id="echoid-s9605" xml:space="preserve"/>
</p>
<div xml:id="echoid-div245" type="float" level="2" n="24">
<note position="left" xlink:label="note-0220-03" xlink:href="note-0220-03a" xml:space="preserve">Fig. 33.</note>
</div>
<p>
  <s xml:id="echoid-s9606" xml:space="preserve">Nam per N, O ducantur ipſi curvæ perpendiculares EN, FO; </s>
  <s xml:id="echoid-s9607" xml:space="preserve">hæ <lb/>cum ipſa HM intra curvam convenient, puta ad R, &amp; </s>
  <s xml:id="echoid-s9608" xml:space="preserve">P; </s>
  <s xml:id="echoid-s9609" xml:space="preserve">ſibi verò <lb/>ipſis in Q.</s>
  <s xml:id="echoid-s9610" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9611" xml:space="preserve">Liquet jam eſſe ang. </s>
  <s xml:id="echoid-s9612" xml:space="preserve">FOK = ang, FPH &amp;</s>
  <s xml:id="echoid-s9613" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9614" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s9615" xml:space="preserve">PRQ = ang. <lb/></s>
  <s xml:id="echoid-s9616" xml:space="preserve">NRH = ang ENJ. </s>
  <s xml:id="echoid-s9617" xml:space="preserve">Cùm ergò ſit ang. </s>
  <s xml:id="echoid-s9618" xml:space="preserve">FOK major angulo ENJ, <lb/>liquet propoſitum.</s>
  <s xml:id="echoid-s9619" xml:space="preserve"/>
</p>
<pb o="43" file="0221" n="236" rhead=""/>
<p>
  <s xml:id="echoid-s9620" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s9621" xml:space="preserve">Si à puncto quopiam Hin perpendiculari HM aſſumpto <lb/>ducantur ad curvam rectæ HN, HO; </s>
  <s xml:id="echoid-s9622" xml:space="preserve">harum propior HN, remoti-<lb/>ore HO rectiùs incidet.</s>
  <s xml:id="echoid-s9623" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9624" xml:space="preserve">Nam ducantur EN, FO curvæ perpendiculares, &amp; </s>
  <s xml:id="echoid-s9625" xml:space="preserve">IN, KO ad <lb/>
<anchor type="note" xlink:label="note-0221-01a" xlink:href="note-0221-01"/>
ipſam HM parallelæ. </s>
  <s xml:id="echoid-s9626" xml:space="preserve">Eſt igitur ang. </s>
  <s xml:id="echoid-s9627" xml:space="preserve">FOK &amp;</s>
  <s xml:id="echoid-s9628" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9629" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s9630" xml:space="preserve">ENI. </s>
  <s xml:id="echoid-s9631" xml:space="preserve">Item <lb/>ang. </s>
  <s xml:id="echoid-s9632" xml:space="preserve">OHM &amp;</s>
  <s xml:id="echoid-s9633" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9634" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s9635" xml:space="preserve">NHM. </s>
  <s xml:id="echoid-s9636" xml:space="preserve">hoc eſt ang. </s>
  <s xml:id="echoid-s9637" xml:space="preserve">KOH &amp;</s>
  <s xml:id="echoid-s9638" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9639" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s9640" xml:space="preserve">INH. <lb/></s>
  <s xml:id="echoid-s9641" xml:space="preserve">quare ang. </s>
  <s xml:id="echoid-s9642" xml:space="preserve">FOK + KOH &amp;</s>
  <s xml:id="echoid-s9643" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9644" xml:space="preserve">ang ENI + INH. </s>
  <s xml:id="echoid-s9645" xml:space="preserve">hoc eſt ang. </s>
  <s xml:id="echoid-s9646" xml:space="preserve"><lb/>FOH &amp;</s>
  <s xml:id="echoid-s9647" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s9648" xml:space="preserve">ang. </s>
  <s xml:id="echoid-s9649" xml:space="preserve">ENH. </s>
  <s xml:id="echoid-s9650" xml:space="preserve">Unde conſtat Propoſitum.</s>
  <s xml:id="echoid-s9651" xml:space="preserve"/>
</p>
<div xml:id="echoid-div246" type="float" level="2" n="25">
<note position="right" xlink:label="note-0221-01" xlink:href="note-0221-01a" xml:space="preserve">Fig. 34.</note>
</div>
<p>
  <s xml:id="echoid-s9652" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s9653" xml:space="preserve">Hinc patet à perpendiculari progrediendo, (ab uno <lb/>nempe puncto H) iucidentium _obliquitatem_ creſcere, donec ad illam <lb/>devenitur, quæ _curvam_ tangit, omnium obliquiſſima.</s>
  <s xml:id="echoid-s9654" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9655" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s9656" xml:space="preserve">Porrò ſi introrſum jam ſumatur punctum H, &amp; </s>
  <s xml:id="echoid-s9657" xml:space="preserve">ab eo in-<lb/>
<anchor type="note" xlink:label="note-0221-02a" xlink:href="note-0221-02"/>
cidens HM ſit omnium curvæ incidentium minima; </s>
  <s xml:id="echoid-s9658" xml:space="preserve">erit HM _curvæ_ <lb/>perpendicularis, ſeu tangenti MT.</s>
  <s xml:id="echoid-s9659" xml:space="preserve"/>
</p>
<div xml:id="echoid-div247" type="float" level="2" n="26">
<note position="right" xlink:label="note-0221-02" xlink:href="note-0221-02a" xml:space="preserve">Fig. 35.</note>
</div>
<p>
  <s xml:id="echoid-s9660" xml:space="preserve">Nam dicaliam MR tangenti perpendicularem eſſe. </s>
  <s xml:id="echoid-s9661" xml:space="preserve">ergò HR &amp;</s>
  <s xml:id="echoid-s9662" xml:space="preserve">lt; <lb/></s>
  <s xml:id="echoid-s9663" xml:space="preserve">
<anchor type="note" xlink:label="note-0221-03a" xlink:href="note-0221-03"/>
HM. </s>
  <s xml:id="echoid-s9664" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9665" xml:space="preserve">magìs HO &amp;</s>
  <s xml:id="echoid-s9666" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9667" xml:space="preserve">HM. </s>
  <s xml:id="echoid-s9668" xml:space="preserve">quare HM non eſt minima contra <lb/>_Hypotheſin_.</s>
  <s xml:id="echoid-s9669" xml:space="preserve"/>
</p>
<div xml:id="echoid-div248" type="float" level="2" n="27">
<note position="right" xlink:label="note-0221-03" xlink:href="note-0221-03a" xml:space="preserve">_Apoll. V._ 32.</note>
</div>
<p>
  <s xml:id="echoid-s9670" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s9671" xml:space="preserve">Item ſi recta HM ſit omnium ab H curvæ incidentium _maxima_, <lb/>
<anchor type="note" xlink:label="note-0221-04a" xlink:href="note-0221-04"/>
erit HM curvæ perpendicularis.</s>
  <s xml:id="echoid-s9672" xml:space="preserve"/>
</p>
<div xml:id="echoid-div249" type="float" level="2" n="28">
<note position="right" xlink:label="note-0221-04" xlink:href="note-0221-04a" xml:space="preserve">_Apoll. V._ 29.</note>
</div>
<p>
  <s xml:id="echoid-s9673" xml:space="preserve">Nam Circulus Centro H per M deſcriptus extra curvam totus ca-<lb/>
<anchor type="note" xlink:label="note-0221-05a" xlink:href="note-0221-05"/>
det. </s>
  <s xml:id="echoid-s9674" xml:space="preserve">ergò ſi recta MT Circulum tangat, hæc magìs extra curvam <lb/>cadet, eámq; </s>
  <s xml:id="echoid-s9675" xml:space="preserve">proinde continget. </s>
  <s xml:id="echoid-s9676" xml:space="preserve">Eſt autem ang. </s>
  <s xml:id="echoid-s9677" xml:space="preserve">HMT rectus. </s>
  <s xml:id="echoid-s9678" xml:space="preserve">er-<lb/>gò liquet.</s>
  <s xml:id="echoid-s9679" xml:space="preserve"/>
</p>
<div xml:id="echoid-div250" type="float" level="2" n="29">
<note position="right" xlink:label="note-0221-05" xlink:href="note-0221-05a" xml:space="preserve">Fig. 36.</note>
</div>
<p>
  <s xml:id="echoid-s9680" xml:space="preserve">XXI. </s>
  <s xml:id="echoid-s9681" xml:space="preserve">Hinc ſi MT ſit minimæ vel maximæ HM perpendicularis; <lb/></s>
  <s xml:id="echoid-s9682" xml:space="preserve">
<anchor type="note" xlink:label="note-0221-06a" xlink:href="note-0221-06"/>
hæc _curvam_ tanget.</s>
  <s xml:id="echoid-s9683" xml:space="preserve"/>
</p>
<div xml:id="echoid-div251" type="float" level="2" n="30">
<note position="right" xlink:label="note-0221-06" xlink:href="note-0221-06a" xml:space="preserve">_Apoll. V._ 30, 39,</note>
</div>
<p>
  <s xml:id="echoid-s9684" xml:space="preserve">Nam ſi dicatur alia MX tangere; </s>
  <s xml:id="echoid-s9685" xml:space="preserve">erit ideò ang. </s>
  <s xml:id="echoid-s9686" xml:space="preserve">XMH rectus, &amp; </s>
  <s xml:id="echoid-s9687" xml:space="preserve"><lb/>par angulo TMH: </s>
  <s xml:id="echoid-s9688" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s9689" xml:space="preserve">E. </s>
  <s xml:id="echoid-s9690" xml:space="preserve">A.</s>
  <s xml:id="echoid-s9691" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9692" xml:space="preserve">XXII. </s>
  <s xml:id="echoid-s9693" xml:space="preserve">Exhinc ſi recta YM non ſit curvæ perpendicularis; </s>
  <s xml:id="echoid-s9694" xml:space="preserve">in ea <lb/>nulla ſumi poteſt _maxima_, vel _minima._</s>
  <s xml:id="echoid-s9695" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9696" xml:space="preserve">Nam ſi ſumi poſſet, eſſet ex eo ipſo YM curvæ perpendicularis <lb/>
<anchor type="note" xlink:label="note-0221-07a" xlink:href="note-0221-07"/>
contra _Hypotbeſin_.</s>
  <s xml:id="echoid-s9697" xml:space="preserve"/>
</p>
<div xml:id="echoid-div252" type="float" level="2" n="31">
<note position="right" xlink:label="note-0221-07" xlink:href="note-0221-07a" xml:space="preserve">_Apoll. V._ 31, 47.</note>
</div>
<p>
  <s xml:id="echoid-s9698" xml:space="preserve">XXIII. </s>
  <s xml:id="echoid-s9699" xml:space="preserve">Si HM ſit incidentium minima, &amp; </s>
  <s xml:id="echoid-s9700" xml:space="preserve">intra ipſam ſumatur <lb/>
<anchor type="note" xlink:label="note-0221-08a" xlink:href="note-0221-08"/>
punctum quodpiam I; </s>
  <s xml:id="echoid-s9701" xml:space="preserve">erit etiam IM minima.</s>
  <s xml:id="echoid-s9702" xml:space="preserve"/>
</p>
<div xml:id="echoid-div253" type="float" level="2" n="32">
<note position="right" xlink:label="note-0221-08" xlink:href="note-0221-08a" xml:space="preserve">_Apoll. V._ 30.</note>
</div>
<pb o="44" file="0222" n="237" rhead=""/>
<p>
  <s xml:id="echoid-s9703" xml:space="preserve">Cùm enim _circulus centro_ H per M deſcriptus _curvam_ intròrſum <lb/>
<anchor type="note" xlink:label="note-0222-01a" xlink:href="note-0222-01"/>
tangat, etiam magìs _circulus centro_ I _deſcriptus_ introrſum tangat. </s>
  <s xml:id="echoid-s9704" xml:space="preserve">un-<lb/>de liquet.</s>
  <s xml:id="echoid-s9705" xml:space="preserve"/>
</p>
<div xml:id="echoid-div254" type="float" level="2" n="33">
<note position="left" xlink:label="note-0222-01" xlink:href="note-0222-01a" xml:space="preserve">Fig. 37.</note>
</div>
<p>
  <s xml:id="echoid-s9706" xml:space="preserve">XXIV. </s>
  <s xml:id="echoid-s9707" xml:space="preserve">Etiam ſi HM ſit incid<unsure/>entium maxima, &amp; </s>
  <s xml:id="echoid-s9708" xml:space="preserve">extra ipſam accipiatur <lb/>punctum quodpiam I, erit IM maxima.</s>
  <s xml:id="echoid-s9709" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9710" xml:space="preserve">Cùm enim _Circulus Centro_ H _per_ M _deſcriptus curvam_ extrorſum <lb/>contingat, etiam potiori jure _Circulus Centro_ I _per_ M _deſcriptus eandem_ <lb/>
<anchor type="note" xlink:label="note-0222-02a" xlink:href="note-0222-02"/>
_exirorſus continget_. </s>
  <s xml:id="echoid-s9711" xml:space="preserve">unde conſtat _Propoſitum_.</s>
  <s xml:id="echoid-s9712" xml:space="preserve"/>
</p>
<div xml:id="echoid-div255" type="float" level="2" n="34">
<note position="left" xlink:label="note-0222-02" xlink:href="note-0222-02a" xml:space="preserve">_Apoll. V._ 39.</note>
</div>
<p style="it">
  <s xml:id="echoid-s9713" xml:space="preserve">_Cæterùm_ minimarum &amp; </s>
  <s xml:id="echoid-s9714" xml:space="preserve">maximarum propior determinatio pendet <lb/>ex ſpeciali curvæ natura.</s>
  <s xml:id="echoid-s9715" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9716" xml:space="preserve">De hac autem Tabula jam manum auferemus; </s>
  <s xml:id="echoid-s9717" xml:space="preserve">nec enim impræſen-<lb/>tiarum hujuſmodi pleraq; </s>
  <s xml:id="echoid-s9718" xml:space="preserve">complecti profitemur. </s>
  <s xml:id="echoid-s9719" xml:space="preserve">Inſtituto noſtro ſuffi-<lb/>cit hactenus generalis cujuſdam curvarum proprietates comprehenden-<lb/>tis Doctrinæ _ſpecimen_ exhibuiſſe: </s>
  <s xml:id="echoid-s9720" xml:space="preserve">qualis certè, plenior &amp; </s>
  <s xml:id="echoid-s9721" xml:space="preserve">perfectior, <lb/>_baud exiguum videtur rebus Geometricis_ (quæ nempe circa _curvaruns<unsure/>_ <lb/>_proprietates &amp; </s>
  <s xml:id="echoid-s9722" xml:space="preserve">affectiones_ plurimùm occupàntur) _compendiuns allat@-_ <lb/>_ra_. </s>
  <s xml:id="echoid-s9723" xml:space="preserve">Nè dicam culpæ agnatum videri, _Logicæ{q́ue}_ Regulis haud admodum <lb/>congruere, quæ toti cuipiam generi conveniunt, &amp; </s>
  <s xml:id="echoid-s9724" xml:space="preserve">quæ de communi <lb/>quadam origine manant, ea quibuſdam partibus adſcribere, vel ex an-<lb/>guſtiorifonte derivare. </s>
  <s xml:id="echoid-s9725" xml:space="preserve">_Plura_ forſan, &amp; </s>
  <s xml:id="echoid-s9726" xml:space="preserve">_abſtruſiora_ proferemus ali-<lb/>quando. </s>
  <s xml:id="echoid-s9727" xml:space="preserve">Nunc his ſuperſedemus.</s>
  <s xml:id="echoid-s9728" xml:space="preserve"/>
</p>
<pb o="45" file="0223" n="238" rhead=""/>
<p>
  <s xml:id="echoid-s9729" xml:space="preserve">AD eaſdem partes vergentium curvarum, è communi quadam <lb/>generatione deductas, generales aliquot affectiones jam antea <lb/>dudum expoſui; </s>
  <s xml:id="echoid-s9730" xml:space="preserve">illas præſertim, quas à veteribus _Geometris_ obſer-<lb/>varam ſpecialibus, quas ipſi tractant, curvis applicari. </s>
  <s xml:id="echoid-s9731" xml:space="preserve">Jam non <lb/>ingratum facturus videor, ſi complures alias (abſtruſiores quidem <lb/>illas, at non injucundas prorſus, aut inutiles) appoſuero; </s>
  <s xml:id="echoid-s9732" xml:space="preserve">pro mèo <lb/>more quàm conciciſſimè demonſtratas, eâ tamen ratione quoad po-<lb/>tero, quæ cumprimis ſcientifica videtur, hoc eſt quæ nedum con-<lb/>cluſionum veritatem aſſerit, at fontes etiam aperit, unde illa pro-<lb/>manat. </s>
  <s xml:id="echoid-s9733" xml:space="preserve">Verſantur autem præcipuè quæ proferemus, partim _circa_ <lb/>_tangentium abſque calculi moleſtia vel faſtidio inveſtigationem ſi-_ <lb/>_mul ac demonſtrationem expeditam_ (è ſimplicioribus nempe vulga-<lb/>tioribúſque perplexiora minúsque perſpecta deducendo) _partim cir-_ <lb/>_ca mnltarum magnitudinum dimenſiones, tangentium deſignatarum o-_ <lb/>_pe, quam promptiſſimè determinandas_; </s>
  <s xml:id="echoid-s9734" xml:space="preserve">quæ materiæ cùm præ Ge-<lb/>ometricis aliis quodammodò difficiles videntur, tum non penitus <lb/>adhuc (ſicut aliæ quædam) occupatæ vel exhauſtæ ſunt, ad hunc <lb/>ſaltem modum quod ſciam nondum tractatæ Quin è veſtigiorem aggre-<lb/>dimur, _Lemmatica_ quædam utcunque, quorum in reliquis clariùs &amp; </s>
  <s xml:id="echoid-s9735" xml:space="preserve"><lb/>breviùs oſtendendis aliquem proſpicimus uſum, prælibantes.</s>
  <s xml:id="echoid-s9736" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9737" xml:space="preserve">II Sit _angulus rectilineus_ ABC, &amp; </s>
  <s xml:id="echoid-s9738" xml:space="preserve">datum punctum D, ſit i-<lb/>
<anchor type="note" xlink:label="note-0223-01a" xlink:href="note-0223-01"/>
tem linea ODO talis, ut per D ductâ quâvis rectâ DN; </s>
  <s xml:id="echoid-s9739" xml:space="preserve">ſit an-<lb/>guli lateribus intercepta MN æqualis à puncto D, &amp; </s>
  <s xml:id="echoid-s9740" xml:space="preserve">linea ODO <lb/>interceptæ DO; </s>
  <s xml:id="echoid-s9741" xml:space="preserve">erit linea ODO _Hyperbola._</s>
  <s xml:id="echoid-s9742" xml:space="preserve"/>
</p>
<div xml:id="echoid-div256" type="float" level="2" n="35">
<note position="right" xlink:label="note-0223-01" xlink:href="note-0223-01a" xml:space="preserve">Fig. 38.</note>
</div>
<p>
  <s xml:id="echoid-s9743" xml:space="preserve">Nam ducatur DL ad CB parallela occurrénſq; </s>
  <s xml:id="echoid-s9744" xml:space="preserve">ipſi AB in L; </s>
  <s xml:id="echoid-s9745" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9746" xml:space="preserve"><lb/>in protracta BL ſumatur LZ = LB; </s>
  <s xml:id="echoid-s9747" xml:space="preserve">ducatúrq; </s>
  <s xml:id="echoid-s9748" xml:space="preserve">ZS ad BC paral-<lb/>lela; </s>
  <s xml:id="echoid-s9749" xml:space="preserve">item ducatur OK ad BZ parallela. </s>
  <s xml:id="echoid-s9750" xml:space="preserve">Et ob poſitam DO = MN; <lb/></s>
  <s xml:id="echoid-s9751" xml:space="preserve">erit HO = BN; </s>
  <s xml:id="echoid-s9752" xml:space="preserve">ergò quum ſit DH. </s>
  <s xml:id="echoid-s9753" xml:space="preserve">HO:</s>
  <s xml:id="echoid-s9754" xml:space="preserve">: (DL. </s>
  <s xml:id="echoid-s9755" xml:space="preserve">LN:</s>
  <s xml:id="echoid-s9756" xml:space="preserve">: DL-
<pb o="46" file="0224" n="239" rhead=""/>
DH. </s>
  <s xml:id="echoid-s9757" xml:space="preserve">LN - HO:</s>
  <s xml:id="echoid-s9758" xml:space="preserve">: LH. </s>
  <s xml:id="echoid-s9759" xml:space="preserve">LB:</s>
  <s xml:id="echoid-s9760" xml:space="preserve">:) LH. </s>
  <s xml:id="echoid-s9761" xml:space="preserve">HK. </s>
  <s xml:id="echoid-s9762" xml:space="preserve">erit DH x HK = <lb/>HO x LH; </s>
  <s xml:id="echoid-s9763" xml:space="preserve">hoc eſt DL x HK - LH x HK = KO x LH - HK <lb/>x LH. </s>
  <s xml:id="echoid-s9764" xml:space="preserve">unde erit DL x HK = KO x LH. </s>
  <s xml:id="echoid-s9765" xml:space="preserve">vel ZL x LD = ZK <lb/>x KO. </s>
  <s xml:id="echoid-s9766" xml:space="preserve">ergò conſtat lineam ODO eſſe _Hyperbolen_, cujus _Aſymptoti_ <lb/>ZA, ZS. </s>
  <s xml:id="echoid-s9767" xml:space="preserve">Breviùs hoc oſtendi poſſet, producendo rectam NDS. <lb/></s>
  <s xml:id="echoid-s9768" xml:space="preserve">Nam eſt DS = DM = DO ± OM = NM ± OM = ON. </s>
  <s xml:id="echoid-s9769" xml:space="preserve">Simi-<lb/>ter quartam &amp; </s>
  <s xml:id="echoid-s9770" xml:space="preserve">nonam breviùs demonſtres licet.</s>
  <s xml:id="echoid-s9771" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9772" xml:space="preserve">Quinimò ſi MN ad DO quamvis eandem perpetuò rationem pona-<lb/>
<anchor type="note" xlink:label="note-0224-01a" xlink:href="note-0224-01"/>
tur habere (puta datam R ad S) etiam linea ODO _Hyperbola_ erit; <lb/></s>
  <s xml:id="echoid-s9773" xml:space="preserve">Nempe ſi tum fiat R. </s>
  <s xml:id="echoid-s9774" xml:space="preserve">S:</s>
  <s xml:id="echoid-s9775" xml:space="preserve">: LB. </s>
  <s xml:id="echoid-s9776" xml:space="preserve">LZ; </s>
  <s xml:id="echoid-s9777" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9778" xml:space="preserve">R. </s>
  <s xml:id="echoid-s9779" xml:space="preserve">S:</s>
  <s xml:id="echoid-s9780" xml:space="preserve">: DL. </s>
  <s xml:id="echoid-s9781" xml:space="preserve">DE; </s>
  <s xml:id="echoid-s9782" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9783" xml:space="preserve">per <lb/>Z ducatur ZS ad BC; </s>
  <s xml:id="echoid-s9784" xml:space="preserve">ac per E tranſeat RE ad ZA parallela, cum <lb/>ZS conveniens in Y; </s>
  <s xml:id="echoid-s9785" xml:space="preserve">erunt YR, YS dictæ _Hyperbolæ aſymptoti_ <lb/>quod jam ſufficerit innuiſſe.</s>
  <s xml:id="echoid-s9786" xml:space="preserve"/>
</p>
<div xml:id="echoid-div257" type="float" level="2" n="36">
<note position="left" xlink:label="note-0224-01" xlink:href="note-0224-01a" xml:space="preserve">Fig. 38.</note>
</div>
<p>
  <s xml:id="echoid-s9787" xml:space="preserve">Hinc in tranſcurſu noto facilè confici _Problema (quo problematum_ <lb/>_confectiones iſtæ Arcbimedeæ, ac Vieteæ ope primæ Conchoidis peractæ_, <lb/>_ad Sectiones conicas rediguntur_) Per datum punctum D rectam lineam <lb/>ducere, ſic ut anguli dati ABC lateribus intercepta ductæ rectæ pars <lb/>æquetur datæ T. </s>
  <s xml:id="echoid-s9788" xml:space="preserve">Nam deſcriptâ hyperbolâ ODO; </s>
  <s xml:id="echoid-s9789" xml:space="preserve">centro D, in-<lb/>tervallo datam T æquante deſcribatur circulus POQ _hyperbolam_ in-<lb/>terſecans in O; </s>
  <s xml:id="echoid-s9790" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9791" xml:space="preserve">producatur DON; </s>
  <s xml:id="echoid-s9792" xml:space="preserve">fiétq; </s>
  <s xml:id="echoid-s9793" xml:space="preserve">MN = DO = T. <lb/></s>
  <s xml:id="echoid-s9794" xml:space="preserve">Modus autem hic generalior eſt, &amp; </s>
  <s xml:id="echoid-s9795" xml:space="preserve">concinnior eo, quem in _Opticis_ <lb/>tradidimus.</s>
  <s xml:id="echoid-s9796" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9797" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s9798" xml:space="preserve">Sit angulus ABC, et punctum datum D; </s>
  <s xml:id="echoid-s9799" xml:space="preserve">ſit etiam linea O <lb/>
<anchor type="note" xlink:label="note-0224-02a" xlink:href="note-0224-02"/>
BO talis, ut per D ductâ quâpiam rectâ DN, ſit anguli late-<lb/>ribus intercepta MN ad rectâ BC curvâque OBO interceptam <lb/>MO in eadem ſemper ratione (puta X ad Y;) </s>
  <s xml:id="echoid-s9800" xml:space="preserve">erit linea OBO <lb/>_hyperbola_.</s>
  <s xml:id="echoid-s9801" xml:space="preserve"/>
</p>
<div xml:id="echoid-div258" type="float" level="2" n="37">
<note position="left" xlink:label="note-0224-02" xlink:href="note-0224-02a" xml:space="preserve">Fig. 39.</note>
</div>
<p>
  <s xml:id="echoid-s9802" xml:space="preserve">Ducatur enim recta DL ad CB parallela, ipſi AB occurrens <lb/>
<anchor type="note" xlink:label="note-0224-03a" xlink:href="note-0224-03"/>
in L; </s>
  <s xml:id="echoid-s9803" xml:space="preserve">ſecentúrque DL, BL punctis E, F, ut ſit DL. </s>
  <s xml:id="echoid-s9804" xml:space="preserve">DE:</s>
  <s xml:id="echoid-s9805" xml:space="preserve">: X. <lb/></s>
  <s xml:id="echoid-s9806" xml:space="preserve">Y:</s>
  <s xml:id="echoid-s9807" xml:space="preserve">: BL. </s>
  <s xml:id="echoid-s9808" xml:space="preserve">BF; </s>
  <s xml:id="echoid-s9809" xml:space="preserve">tum per E ducatur recta ER, ad BA; </s>
  <s xml:id="echoid-s9810" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9811" xml:space="preserve">per <lb/>F recta FS ad BC parallela; </s>
  <s xml:id="echoid-s9812" xml:space="preserve">concurrántque rectæ ER, FS pun-<lb/>cto Z; </s>
  <s xml:id="echoid-s9813" xml:space="preserve">denuò per punctum O ducatur OH ad AB parallela. </s>
  <s xml:id="echoid-s9814" xml:space="preserve">Jam <lb/>ob DL. </s>
  <s xml:id="echoid-s9815" xml:space="preserve">DH:</s>
  <s xml:id="echoid-s9816" xml:space="preserve">: LN. </s>
  <s xml:id="echoid-s9817" xml:space="preserve">HO:</s>
  <s xml:id="echoid-s9818" xml:space="preserve">: LB + BN. </s>
  <s xml:id="echoid-s9819" xml:space="preserve">HO:</s>
  <s xml:id="echoid-s9820" xml:space="preserve">: DE x LB <lb/>+ DE x BN. </s>
  <s xml:id="echoid-s9821" xml:space="preserve">DE x HO. </s>
  <s xml:id="echoid-s9822" xml:space="preserve">item DL x KO = DE x BN <lb/>(nam DL. </s>
  <s xml:id="echoid-s9823" xml:space="preserve">DE:</s>
  <s xml:id="echoid-s9824" xml:space="preserve">: MN. </s>
  <s xml:id="echoid-s9825" xml:space="preserve">MO:</s>
  <s xml:id="echoid-s9826" xml:space="preserve">: BN. </s>
  <s xml:id="echoid-s9827" xml:space="preserve">KO) &amp; </s>
  <s xml:id="echoid-s9828" xml:space="preserve">DE x LB = DL <lb/>x BF (ob DE. </s>
  <s xml:id="echoid-s9829" xml:space="preserve">DL:</s>
  <s xml:id="echoid-s9830" xml:space="preserve">: BF. </s>
  <s xml:id="echoid-s9831" xml:space="preserve">LB;) </s>
  <s xml:id="echoid-s9832" xml:space="preserve">erit DL. </s>
  <s xml:id="echoid-s9833" xml:space="preserve">DH:</s>
  <s xml:id="echoid-s9834" xml:space="preserve">: DL x BF <lb/>+ DL x KO. </s>
  <s xml:id="echoid-s9835" xml:space="preserve">DE x HO; </s>
  <s xml:id="echoid-s9836" xml:space="preserve">hoc eſt DL x BF + DL x <lb/>KO. </s>
  <s xml:id="echoid-s9837" xml:space="preserve">DH x BF + DH x KO:</s>
  <s xml:id="echoid-s9838" xml:space="preserve">: DL x BF x DL x KO.</s>
  <s xml:id="echoid-s9839" xml:space="preserve">
<pb o="47" file="0225" n="240" rhead=""/>
DE x HO ergò DH x BF + DH x KO = DE x HO; </s>
  <s xml:id="echoid-s9840" xml:space="preserve">hoc eſt <lb/>DH x BF + DH x HO - DH x BL = DE x HO; </s>
  <s xml:id="echoid-s9841" xml:space="preserve">tranſpo-<lb/>nendo igitur eſt DH x HO - DE x HO = DH x BL - DH x <lb/>
<anchor type="note" xlink:label="note-0225-01a" xlink:href="note-0225-01"/>
BF. </s>
  <s xml:id="echoid-s9842" xml:space="preserve">hoc eſt EH x HO = DH x FL; </s>
  <s xml:id="echoid-s9843" xml:space="preserve">vel EH x GO + EH x <lb/>HG = DE x FL + EH x FL; </s>
  <s xml:id="echoid-s9844" xml:space="preserve">quare, demptis æqualibus, eſt EH <lb/>x GO = DE x FL; </s>
  <s xml:id="echoid-s9845" xml:space="preserve">vel ZG x GO = DE x FL; </s>
  <s xml:id="echoid-s9846" xml:space="preserve">cum itaque <lb/>DE x FL ſit quid determinatum, conſtat lineam OBO effe hy-<lb/>perbolam, cujus aſymptoti ZR, ZS.</s>
  <s xml:id="echoid-s9847" xml:space="preserve"/>
</p>
<div xml:id="echoid-div259" type="float" level="2" n="38">
<note position="left" xlink:label="note-0224-03" xlink:href="note-0224-03a" xml:space="preserve">Fig. 39.</note>
<note position="right" xlink:label="note-0225-01" xlink:href="note-0225-01a" xml:space="preserve">Fig. 39.</note>
</div>
<p>
  <s xml:id="echoid-s9848" xml:space="preserve">V. </s>
  <s xml:id="echoid-s9849" xml:space="preserve">Si MO capiatur ad alteras rectæ BC partes, etiam DE. <lb/></s>
  <s xml:id="echoid-s9850" xml:space="preserve">BF ad alteras punctorum D, B partes accipi debent; </s>
  <s xml:id="echoid-s9851" xml:space="preserve">uti Schema <lb/>
<anchor type="note" xlink:label="note-0225-02a" xlink:href="note-0225-02"/>
monſtrat; </s>
  <s xml:id="echoid-s9852" xml:space="preserve">nec abludit modus demonſtrandi.</s>
  <s xml:id="echoid-s9853" xml:space="preserve"/>
</p>
<div xml:id="echoid-div260" type="float" level="2" n="39">
<note position="right" xlink:label="note-0225-02" xlink:href="note-0225-02a" xml:space="preserve">Fig. 40.</note>
</div>
<p>
  <s xml:id="echoid-s9854" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s9855" xml:space="preserve">Conſectarium. </s>
  <s xml:id="echoid-s9856" xml:space="preserve">Si recta BQ angulum ABC ſecet, pér-<lb/>que punctum D ducantur utcunque duæ rectæ MN, XY rectam <lb/>
<anchor type="note" xlink:label="note-0225-03a" xlink:href="note-0225-03"/>
BQ interſecantes punctis OP (quorum utique ſit O propius ip-<lb/>ſi B) erit MN. </s>
  <s xml:id="echoid-s9857" xml:space="preserve">MO &amp;</s>
  <s xml:id="echoid-s9858" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9859" xml:space="preserve">XY. </s>
  <s xml:id="echoid-s9860" xml:space="preserve">XP. </s>
  <s xml:id="echoid-s9861" xml:space="preserve">Nam per O deſcripta con-<lb/>cipiatur _hyperbola_ VOB (qualem jam mox attigimus, ſic ut inter-<lb/>ceptæ rationem habeant illam quam MN ad MO) erit igitur <lb/>MN. </s>
  <s xml:id="echoid-s9862" xml:space="preserve">MO:</s>
  <s xml:id="echoid-s9863" xml:space="preserve">: (XY. </s>
  <s xml:id="echoid-s9864" xml:space="preserve">XV) &amp;</s>
  <s xml:id="echoid-s9865" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9866" xml:space="preserve">XY. </s>
  <s xml:id="echoid-s9867" xml:space="preserve">XP.</s>
  <s xml:id="echoid-s9868" xml:space="preserve"/>
</p>
<div xml:id="echoid-div261" type="float" level="2" n="40">
<note position="right" xlink:label="note-0225-03" xlink:href="note-0225-03a" xml:space="preserve">Fig. 41.</note>
</div>
<p>
  <s xml:id="echoid-s9869" xml:space="preserve">_Coroll._ </s>
  <s xml:id="echoid-s9870" xml:space="preserve">Dividendo eſt NO. </s>
  <s xml:id="echoid-s9871" xml:space="preserve">MO &amp;</s>
  <s xml:id="echoid-s9872" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9873" xml:space="preserve">YP. </s>
  <s xml:id="echoid-s9874" xml:space="preserve">PX.</s>
  <s xml:id="echoid-s9875" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9876" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s9877" xml:space="preserve">Quinimò ſi plures lineæ BQ, BG angulum ABC ſecent; <lb/></s>
  <s xml:id="echoid-s9878" xml:space="preserve">
<anchor type="note" xlink:label="note-0225-04a" xlink:href="note-0225-04"/>
&amp; </s>
  <s xml:id="echoid-s9879" xml:space="preserve">à puncto D projiciantur rectæ DN, DY (quæ rectas alteras <lb/>interſecant ut vides; </s>
  <s xml:id="echoid-s9880" xml:space="preserve">quarúmque DN puncto B vicinior;) </s>
  <s xml:id="echoid-s9881" xml:space="preserve">erit <lb/>NE. </s>
  <s xml:id="echoid-s9882" xml:space="preserve">MO &amp;</s>
  <s xml:id="echoid-s9883" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9884" xml:space="preserve">YF. </s>
  <s xml:id="echoid-s9885" xml:space="preserve">VX.</s>
  <s xml:id="echoid-s9886" xml:space="preserve"/>
</p>
<div xml:id="echoid-div262" type="float" level="2" n="41">
<note position="right" xlink:label="note-0225-04" xlink:href="note-0225-04a" xml:space="preserve">Fig. 42.</note>
</div>
<p>
  <s xml:id="echoid-s9887" xml:space="preserve">Nam NE. </s>
  <s xml:id="echoid-s9888" xml:space="preserve">EO &amp;</s>
  <s xml:id="echoid-s9889" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9890" xml:space="preserve">YF. </s>
  <s xml:id="echoid-s9891" xml:space="preserve">FV; </s>
  <s xml:id="echoid-s9892" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9893" xml:space="preserve">EO. </s>
  <s xml:id="echoid-s9894" xml:space="preserve">OM &amp;</s>
  <s xml:id="echoid-s9895" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9896" xml:space="preserve">FV. </s>
  <s xml:id="echoid-s9897" xml:space="preserve">VX. </s>
  <s xml:id="echoid-s9898" xml:space="preserve">i-<lb/>gitur ex æquo eſt NE. </s>
  <s xml:id="echoid-s9899" xml:space="preserve">OM &amp;</s>
  <s xml:id="echoid-s9900" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s9901" xml:space="preserve">YF. </s>
  <s xml:id="echoid-s9902" xml:space="preserve">VX.</s>
  <s xml:id="echoid-s9903" xml:space="preserve">‖</s>
</p>
<p>
  <s xml:id="echoid-s9904" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s9905" xml:space="preserve">Etiam exindè patet, per B (ad partes alterutras) rectam <lb/>duci poſſe; </s>
  <s xml:id="echoid-s9906" xml:space="preserve">ità ut è D eductarum partes ab illa rectáque BC ad <lb/>interceptas à rectis BA, BC rationem habeant minorem quâpi-<lb/>am datâ.</s>
  <s xml:id="echoid-s9907" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9908" xml:space="preserve">Nam ſumatur PQ = PZ; </s>
  <s xml:id="echoid-s9909" xml:space="preserve">ergò connexa BQ _hyperbolam_ O <lb/>BO tangit; </s>
  <s xml:id="echoid-s9910" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9911" xml:space="preserve">liquet à rectis BQ, BC interceptas ad intercep-<lb/>tas à BC, BA minorem rationem habere, quàm habent inter-<lb/>ceptæ ab hyperbolâ OBO &amp; </s>
  <s xml:id="echoid-s9912" xml:space="preserve">recta BC ad eaſdem; </s>
  <s xml:id="echoid-s9913" xml:space="preserve">hoc eſt mi-<lb/>norem datâ quâpiam.</s>
  <s xml:id="echoid-s9914" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9915" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s9916" xml:space="preserve">Sit rurſum angulus rectilineus ABC, &amp; </s>
  <s xml:id="echoid-s9917" xml:space="preserve">punctum D; </s>
  <s xml:id="echoid-s9918" xml:space="preserve">item <lb/>
<anchor type="note" xlink:label="note-0225-05a" xlink:href="note-0225-05"/>
<pb o="48" file="0226" n="241" rhead=""/>
linea OOO talis, ut ſi è D utcunque ducatur recta DO, ſecans <lb/>anguli latera punctis M, N, habeat DM ad NO ſemper eandem <lb/>
<anchor type="note" xlink:label="note-0226-01a" xlink:href="note-0226-01"/>
rationem (puta X ad Y) erit etiam linea OOO hyperbola.</s>
  <s xml:id="echoid-s9919" xml:space="preserve"/>
</p>
<div xml:id="echoid-div263" type="float" level="2" n="42">
<note position="right" xlink:label="note-0225-05" xlink:href="note-0225-05a" xml:space="preserve">Fig. 43.</note>
<note position="left" xlink:label="note-0226-01" xlink:href="note-0226-01a" xml:space="preserve">Fig. 43.</note>
</div>
<p>
  <s xml:id="echoid-s9920" xml:space="preserve">Nam ducatur DL ad BC parallela; </s>
  <s xml:id="echoid-s9921" xml:space="preserve">ſitque DL. </s>
  <s xml:id="echoid-s9922" xml:space="preserve">DE:</s>
  <s xml:id="echoid-s9923" xml:space="preserve">: X. </s>
  <s xml:id="echoid-s9924" xml:space="preserve">Y; <lb/></s>
  <s xml:id="echoid-s9925" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9926" xml:space="preserve">per E ducatur ER ad AB parallela; </s>
  <s xml:id="echoid-s9927" xml:space="preserve">ſecans BC in Z; </s>
  <s xml:id="echoid-s9928" xml:space="preserve">de-<lb/>mum per O ducatur OH ad BA parallela.</s>
  <s xml:id="echoid-s9929" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9930" xml:space="preserve">Eſt jam DL. </s>
  <s xml:id="echoid-s9931" xml:space="preserve">DE:</s>
  <s xml:id="echoid-s9932" xml:space="preserve">: DM. </s>
  <s xml:id="echoid-s9933" xml:space="preserve">NO:</s>
  <s xml:id="echoid-s9934" xml:space="preserve">: LM. </s>
  <s xml:id="echoid-s9935" xml:space="preserve">GO (ob ſimilia tri-<lb/>angula DLM, NGO):</s>
  <s xml:id="echoid-s9936" xml:space="preserve">: LM x DH. </s>
  <s xml:id="echoid-s9937" xml:space="preserve">GO x DH item DL x <lb/>HO = LM x DH (ob DL. </s>
  <s xml:id="echoid-s9938" xml:space="preserve">LM:</s>
  <s xml:id="echoid-s9939" xml:space="preserve">: DH. </s>
  <s xml:id="echoid-s9940" xml:space="preserve">HO) quare DL. </s>
  <s xml:id="echoid-s9941" xml:space="preserve">DE:</s>
  <s xml:id="echoid-s9942" xml:space="preserve">: <lb/>DL x HO. </s>
  <s xml:id="echoid-s9943" xml:space="preserve">GO x DH hoc eſt DL x HO. </s>
  <s xml:id="echoid-s9944" xml:space="preserve">DE x HO:</s>
  <s xml:id="echoid-s9945" xml:space="preserve">: DL x HO. <lb/></s>
  <s xml:id="echoid-s9946" xml:space="preserve">GO x DH adeóq; </s>
  <s xml:id="echoid-s9947" xml:space="preserve">DE x HO = GO x DH. </s>
  <s xml:id="echoid-s9948" xml:space="preserve">hoc eſt DE x HG + DE x <lb/>GO = GO x DE + GO x EH quare (communi ſublato) eſt <lb/>DE x HG = GO x EH; </s>
  <s xml:id="echoid-s9949" xml:space="preserve">ſeu DE x HG = GO x ZG. </s>
  <s xml:id="echoid-s9950" xml:space="preserve">Pa-<lb/>tet itaque curvam OOO eſſe _hyperbolam_ cujus _aſymptoti_ ZR <lb/>ZC.</s>
  <s xml:id="echoid-s9951" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9952" xml:space="preserve">_Coroll_. </s>
  <s xml:id="echoid-s9953" xml:space="preserve">Si ratio data ſit æqualitatis (ceu DM = NO,) ipſæ AB, <lb/>CB aſymptoti erunt.</s>
  <s xml:id="echoid-s9954" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9955" xml:space="preserve">Sequentia quædam, quia magìs id perſpicuum videtur, Alge-<lb/>bricè monſtrabimus.</s>
  <s xml:id="echoid-s9956" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s9957" xml:space="preserve">X. </s>
  <s xml:id="echoid-s9958" xml:space="preserve">Eſto poſitione data recta ID, in qua punctum deſignatum D, <lb/>ſit item curva DNN talis ut in ID ſumpto quopiam puncto G, &amp; </s>
  <s xml:id="echoid-s9959" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0226-02a" xlink:href="note-0226-02"/>
ductâ rectâ GN ad poſitionem datam IK paràllelá; </s>
  <s xml:id="echoid-s9960" xml:space="preserve">tum adſumptis <lb/>determinatis rectis _m, b_; </s>
  <s xml:id="echoid-s9961" xml:space="preserve">poſitiſq; </s>
  <s xml:id="echoid-s9962" xml:space="preserve">DG = _x_, &amp; </s>
  <s xml:id="echoid-s9963" xml:space="preserve">GN = _y_; </s>
  <s xml:id="echoid-s9964" xml:space="preserve">ſit <lb/>conſtantèr _m y_ + _x y_ = {_m_/_b_}_x x_; </s>
  <s xml:id="echoid-s9965" xml:space="preserve">erit linea DNN _hyperbola_; </s>
  <s xml:id="echoid-s9966" xml:space="preserve">quæ <lb/>ſic determinatur; </s>
  <s xml:id="echoid-s9967" xml:space="preserve">ſumantur DM, &amp; </s>
  <s xml:id="echoid-s9968" xml:space="preserve">DO (hinc indè) pares ipſi _m_; <lb/></s>
  <s xml:id="echoid-s9969" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9970" xml:space="preserve">per M ducatur M L@ad IK parallela, factóq; </s>
  <s xml:id="echoid-s9971" xml:space="preserve">_b. </s>
  <s xml:id="echoid-s9972" xml:space="preserve">m_:</s>
  <s xml:id="echoid-s9973" xml:space="preserve">: _m_. </s>
  <s xml:id="echoid-s9974" xml:space="preserve">MQ; </s>
  <s xml:id="echoid-s9975" xml:space="preserve">ſit <lb/>MZ = 2 MQ = {2_mm_;</s>
  <s xml:id="echoid-s9976" xml:space="preserve">/_b_} tum per Z, O traducatur recta ZT; </s>
  <s xml:id="echoid-s9977" xml:space="preserve">erunt <lb/>ZM, ZT aſymptoti.</s>
  <s xml:id="echoid-s9978" xml:space="preserve"/>
</p>
<div xml:id="echoid-div264" type="float" level="2" n="43">
<note position="left" xlink:label="note-0226-02" xlink:href="note-0226-02a" xml:space="preserve">Fig. 44.</note>
</div>
<p>
  <s xml:id="echoid-s9979" xml:space="preserve">Ducatur enim ZS ad MO parallela, cui occurrat N Gin R (quæ <lb/>&amp; </s>
  <s xml:id="echoid-s9980" xml:space="preserve">ipſam ZT ſect in P). </s>
  <s xml:id="echoid-s9981" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s9982" xml:space="preserve">connectatur DQ. </s>
  <s xml:id="echoid-s9983" xml:space="preserve">Eſt ergò PN = RG <lb/>+ GN - RP. </s>
  <s xml:id="echoid-s9984" xml:space="preserve">Verùm eſt MD. </s>
  <s xml:id="echoid-s9985" xml:space="preserve">MQ:</s>
  <s xml:id="echoid-s9986" xml:space="preserve">: ZR (MG). </s>
  <s xml:id="echoid-s9987" xml:space="preserve">RP; </s>
  <s xml:id="echoid-s9988" xml:space="preserve">hoc <lb/>eſt _m_. </s>
  <s xml:id="echoid-s9989" xml:space="preserve">{_mm_/_b_}:</s>
  <s xml:id="echoid-s9990" xml:space="preserve">: _m_ + _x_. </s>
  <s xml:id="echoid-s9991" xml:space="preserve">RP = {_mm_/_b_} + {_mx._</s>
  <s xml:id="echoid-s9992" xml:space="preserve">/_b_} adeóq; </s>
  <s xml:id="echoid-s9993" xml:space="preserve">RG - RP <lb/>= {_mm_/_b_} - {_mx._</s>
  <s xml:id="echoid-s9994" xml:space="preserve">/_b_} ergò PN = {_mm_ - _mx_/_b_} + _y_. </s>
  <s xml:id="echoid-s9995" xml:space="preserve">Unde PN x MG <lb/>= {_m_<emph style="sub">3</emph>/_b_} + _my_ + _xy_ - {_mxx._</s>
  <s xml:id="echoid-s9996" xml:space="preserve">/_b_} Verùm (ex hypotheſi) eſt _m y_
<pb o="49" file="0227" n="242" rhead=""/>
+ _xy_ - {_mxx_/_b_} = _o_. </s>
  <s xml:id="echoid-s9997" xml:space="preserve">ergò PN x MG = {_m_<emph style="sub">3</emph>/_b_} = MD x ZQ. <lb/></s>
  <s xml:id="echoid-s9998" xml:space="preserve">vel PN. </s>
  <s xml:id="echoid-s9999" xml:space="preserve">ZQ:</s>
  <s xml:id="echoid-s10000" xml:space="preserve">: (MD. </s>
  <s xml:id="echoid-s10001" xml:space="preserve">MG:</s>
  <s xml:id="echoid-s10002" xml:space="preserve">:) QD.</s>
  <s xml:id="echoid-s10003" xml:space="preserve">ZP. </s>
  <s xml:id="echoid-s10004" xml:space="preserve">Quapropter eſt <lb/>
<anchor type="note" xlink:label="note-0227-01a" xlink:href="note-0227-01"/>
PN x ZP = ZQ x QD. </s>
  <s xml:id="echoid-s10005" xml:space="preserve">Unde palàm eſt curvam DNN eſſe hy-<lb/>perbolam, cujus aſymptoti ZM, ZT.</s>
  <s xml:id="echoid-s10006" xml:space="preserve"/>
</p>
<div xml:id="echoid-div265" type="float" level="2" n="44">
<note position="right" xlink:label="note-0227-01" xlink:href="note-0227-01a" xml:space="preserve">Fig. 44.</note>
</div>
<p>
  <s xml:id="echoid-s10007" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s10008" xml:space="preserve">Notetur; </s>
  <s xml:id="echoid-s10009" xml:space="preserve">ſi æquatio ſit _my_ - _xy_ = {_m_/_b_}_xx_; </s>
  <s xml:id="echoid-s10010" xml:space="preserve">eadem ha-<lb/>bebitur _hyperbola_; </s>
  <s xml:id="echoid-s10011" xml:space="preserve">tunc ſolùm puncta G ad partes DM ſumuntur. <lb/></s>
  <s xml:id="echoid-s10012" xml:space="preserve">Quin &amp; </s>
  <s xml:id="echoid-s10013" xml:space="preserve">fi æquatio ſit _xy_ - _my_ = {_m_/_b_} _xx_; </s>
  <s xml:id="echoid-s10014" xml:space="preserve">puncta G ultra M <lb/>capiendo, proveniet _hyperbola_, huic ipſi _conjugata_.</s>
  <s xml:id="echoid-s10015" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10016" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s10017" xml:space="preserve">Sit Triangulum BDF; </s>
  <s xml:id="echoid-s10018" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10019" xml:space="preserve">linea DNN talis, ut ductâ ut-<lb/>
<anchor type="note" xlink:label="note-0227-02a" xlink:href="note-0227-02"/>
cunque RN ad BD parallelâ (quæ lineas BF, DF, DNN ſecet <lb/>punctis R, G, N) connexâque rectâ DN; </s>
  <s xml:id="echoid-s10020" xml:space="preserve">ſit perpetuò DN propor-<lb/>tione media inter RN, NG; </s>
  <s xml:id="echoid-s10021" xml:space="preserve">erit linea DNN _hyperbola_.</s>
  <s xml:id="echoid-s10022" xml:space="preserve"/>
</p>
<div xml:id="echoid-div266" type="float" level="2" n="45">
<note position="right" xlink:label="note-0227-02" xlink:href="note-0227-02a" xml:space="preserve">Fig. 45.</note>
</div>
<p>
  <s xml:id="echoid-s10023" xml:space="preserve">Per D ducatur DK ad DB perpendicularis (ſecans ipſam RN in E) <lb/>&amp; </s>
  <s xml:id="echoid-s10024" xml:space="preserve">ſit FH ad DB parallela; </s>
  <s xml:id="echoid-s10025" xml:space="preserve">vocentúrque DB = _b_; </s>
  <s xml:id="echoid-s10026" xml:space="preserve">DF = _d_; </s>
  <s xml:id="echoid-s10027" xml:space="preserve">FH <lb/>= _f_; </s>
  <s xml:id="echoid-s10028" xml:space="preserve">tum DG = _x_; </s>
  <s xml:id="echoid-s10029" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10030" xml:space="preserve">GN = _y_; </s>
  <s xml:id="echoid-s10031" xml:space="preserve">Eſtque _d. </s>
  <s xml:id="echoid-s10032" xml:space="preserve">f:</s>
  <s xml:id="echoid-s10033" xml:space="preserve">: x._ </s>
  <s xml:id="echoid-s10034" xml:space="preserve">{_fx_/_d_} = _GE_; <lb/></s>
  <s xml:id="echoid-s10035" xml:space="preserve">unde {_zfxy_/_d_} + _xx_ + _yy_ = 2 EG x GN + DGq + GNq <lb/>= DN q. </s>
  <s xml:id="echoid-s10036" xml:space="preserve">Porrò eſt _d. </s>
  <s xml:id="echoid-s10037" xml:space="preserve">b_:</s>
  <s xml:id="echoid-s10038" xml:space="preserve">: FG. </s>
  <s xml:id="echoid-s10039" xml:space="preserve">GR:</s>
  <s xml:id="echoid-s10040" xml:space="preserve">: _d_ - _x_.</s>
  <s xml:id="echoid-s10041" xml:space="preserve">RG = _b_ - {_bx._</s>
  <s xml:id="echoid-s10042" xml:space="preserve">/_d_} Un-<lb/>de RN = _b_ - {_bx_/_d_} + _y_. </s>
  <s xml:id="echoid-s10043" xml:space="preserve">Et ideò _by_ - {_bxy_/_d_} + _yy_ = RN x <lb/>NG = DNq = {2_fxy_/_d_} + _xx_ + _yy_. </s>
  <s xml:id="echoid-s10044" xml:space="preserve">quare _by_ - {_bxy_/_d_} = <lb/>{2_fxy_/_d_} + _xx_. </s>
  <s xml:id="echoid-s10045" xml:space="preserve">quam æquationem ordinando fit {_db_/2_f_+_b_}_y_ - _yx_ = <lb/>{_d_/2_f_+_b_}_xx_. </s>
  <s xml:id="echoid-s10046" xml:space="preserve">quòd ſi ponatur _m_ = {_db_/2_f_+_b_<emph style="sub">3</emph>} erit _my_ - _xy_ = <lb/>{_m_/_b_}_xx_. </s>
  <s xml:id="echoid-s10047" xml:space="preserve">Unde liquet DNN eſſe _hyperbolam_, qualis habetur in præ-<lb/>cedente determinata,</s>
</p>
<p>
  <s xml:id="echoid-s10048" xml:space="preserve">Not. </s>
  <s xml:id="echoid-s10049" xml:space="preserve">Siangulus DGN rectus fuerit, evaneſcente tum _f_ = _o_, erit
<pb o="50" file="0228" n="243" rhead=""/>
_d_ = _m_; </s>
  <s xml:id="echoid-s10050" xml:space="preserve">vel _dy_ - _xy_ = {_d_/_b_}_x x_. </s>
  <s xml:id="echoid-s10051" xml:space="preserve">Aliaquædam hîc (nonnulla forſan παρέ<unsure/>ργως) <lb/>inſeremus.</s>
  <s xml:id="echoid-s10052" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10053" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s10054" xml:space="preserve">Sit poſitione data recta ID; </s>
  <s xml:id="echoid-s10055" xml:space="preserve">ſit item curva DNN talis, <lb/>
<anchor type="note" xlink:label="note-0228-01a" xlink:href="note-0228-01"/>
utin ID ſumpto puncto quopiam G, ductâque rectâ GN ad poſitio-<lb/>nem datam IK parallelâ; </s>
  <s xml:id="echoid-s10056" xml:space="preserve">ſumptiſque determinatis lineis _g, m, r_; <lb/></s>
  <s xml:id="echoid-s10057" xml:space="preserve">poſitíſque DG = _x_, &amp; </s>
  <s xml:id="echoid-s10058" xml:space="preserve">GN = _y_; </s>
  <s xml:id="echoid-s10059" xml:space="preserve">ſit perpetim _y x_ + _gx_ - _my_ = <lb/>{_m_/_r_}_x x_; </s>
  <s xml:id="echoid-s10060" xml:space="preserve">linea DNN erit _hyperbola_, ſic determinabilis: </s>
  <s xml:id="echoid-s10061" xml:space="preserve">Sumatur <lb/>DM = _m_; </s>
  <s xml:id="echoid-s10062" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10063" xml:space="preserve">per M ducatur ML ad IK parallela; </s>
  <s xml:id="echoid-s10064" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10065" xml:space="preserve">in hac acci-<lb/>piatur MQ = {_mm_/_r_}; </s>
  <s xml:id="echoid-s10066" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10067" xml:space="preserve">ſit QY = MQ; </s>
  <s xml:id="echoid-s10068" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10069" xml:space="preserve">ab MY auferatur <lb/>YZ = _g_; </s>
  <s xml:id="echoid-s10070" xml:space="preserve">connexâque QD, ducatur ZT ad QD parallelâ; </s>
  <s xml:id="echoid-s10071" xml:space="preserve">erunt <lb/>ZM, ZT _aſymptoti_.</s>
  <s xml:id="echoid-s10072" xml:space="preserve"/>
</p>
<div xml:id="echoid-div267" type="float" level="2" n="46">
<note position="left" xlink:label="note-0228-01" xlink:href="note-0228-01a" xml:space="preserve">Fig. 46.</note>
</div>
<p>
  <s xml:id="echoid-s10073" xml:space="preserve">Nam ducatur ZS ad MD parallela; </s>
  <s xml:id="echoid-s10074" xml:space="preserve">cui occurrat GN producta in <lb/>R (ſed &amp; </s>
  <s xml:id="echoid-s10075" xml:space="preserve">GR ipſam ZT ſecet in P). </s>
  <s xml:id="echoid-s10076" xml:space="preserve">Eſtque jam PN = RG -<lb/>RP - GN = {_mm_/_r_} - _g_ + {_mx_/_r_} - _y_. </s>
  <s xml:id="echoid-s10077" xml:space="preserve">adeoque PN x MG = {_m_<emph style="sub">3</emph>/_r_} <lb/>- _mg_ + _yx_ + _gx_ - _my_ - {_m_/_r_}_x x_ = {_m_<emph style="sub">3</emph>/_r_} - _mg_ + _o_. </s>
  <s xml:id="echoid-s10078" xml:space="preserve">= {_m_<emph style="sub">3</emph>/_r_} <lb/>- _mg_ = DM x ZQ. </s>
  <s xml:id="echoid-s10079" xml:space="preserve">unde PN. </s>
  <s xml:id="echoid-s10080" xml:space="preserve">ZQ:</s>
  <s xml:id="echoid-s10081" xml:space="preserve">: (DM. </s>
  <s xml:id="echoid-s10082" xml:space="preserve">MG:</s>
  <s xml:id="echoid-s10083" xml:space="preserve">:) QD. <lb/></s>
  <s xml:id="echoid-s10084" xml:space="preserve">ZP. </s>
  <s xml:id="echoid-s10085" xml:space="preserve">ergo PN x ZP = ZQ x QD. </s>
  <s xml:id="echoid-s10086" xml:space="preserve">Liquetigitur curvam DNN <lb/>eſſe _hyperbolam_, cujus _aſymptoti_ ZM, ZT.</s>
  <s xml:id="echoid-s10087" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10088" xml:space="preserve">Siæquatiò ſit - _yz_ + _gx_ + _my_ = {_m_/_r_} _xx_; </s>
  <s xml:id="echoid-s10089" xml:space="preserve">eadem erit _hyper-_ <lb/>_bola_. </s>
  <s xml:id="echoid-s10090" xml:space="preserve">Sed puncta G inter B, M tunc accipiuntur; </s>
  <s xml:id="echoid-s10091" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10092" xml:space="preserve">ità prout aliis <lb/>ac aliis locis puncta G deſignantur, æquationis ſigna variantur; </s>
  <s xml:id="echoid-s10093" xml:space="preserve">at <lb/>non eſt ea jam exponendi locus.</s>
  <s xml:id="echoid-s10094" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10095" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s10096" xml:space="preserve">Poſitione datæ ſint rectæ DB, BA; </s>
  <s xml:id="echoid-s10097" xml:space="preserve">pérque rectam DB <lb/>feratur recta CX ad BA parallela; </s>
  <s xml:id="echoid-s10098" xml:space="preserve">item per punctum D rotando <lb/>
<anchor type="note" xlink:label="note-0228-02a" xlink:href="note-0228-02"/>
tranſeat recta DY, ſic ipſam BA ſecans in E, ut ſit inter rectas BE, <lb/>DC eadem ſemper proportio (puta quæ cujuſdam aſſignatæ R ad DB) <lb/>rectæ verò DE, CX ſe interſecent punctis N; </s>
  <s xml:id="echoid-s10099" xml:space="preserve">erit linea DNN _Pa-_ <lb/>_rabola_.</s>
  <s xml:id="echoid-s10100" xml:space="preserve"/>
</p>
<div xml:id="echoid-div268" type="float" level="2" n="47">
<note position="left" xlink:label="note-0228-02" xlink:href="note-0228-02a" xml:space="preserve">Fig. 47.<unsure/></note>
</div>
<p>
  <s xml:id="echoid-s10101" xml:space="preserve">Nam ſit R. </s>
  <s xml:id="echoid-s10102" xml:space="preserve">DB:</s>
  <s xml:id="echoid-s10103" xml:space="preserve">: DB. </s>
  <s xml:id="echoid-s10104" xml:space="preserve">P. </s>
  <s xml:id="echoid-s10105" xml:space="preserve">Eſt ergò BE. </s>
  <s xml:id="echoid-s10106" xml:space="preserve">DC:</s>
  <s xml:id="echoid-s10107" xml:space="preserve">: DB. </s>
  <s xml:id="echoid-s10108" xml:space="preserve">P. </s>
  <s xml:id="echoid-s10109" xml:space="preserve">Item <lb/>eſt DB. </s>
  <s xml:id="echoid-s10110" xml:space="preserve">BE:</s>
  <s xml:id="echoid-s10111" xml:space="preserve">: DC. </s>
  <s xml:id="echoid-s10112" xml:space="preserve">CN. </s>
  <s xml:id="echoid-s10113" xml:space="preserve">ergò DB. </s>
  <s xml:id="echoid-s10114" xml:space="preserve">BE + BE. </s>
  <s xml:id="echoid-s10115" xml:space="preserve">DC = DC.</s>
  <s xml:id="echoid-s10116" xml:space="preserve">
<pb o="51" file="0229" n="244" rhead=""/>
CN + DB. </s>
  <s xml:id="echoid-s10117" xml:space="preserve">P. </s>
  <s xml:id="echoid-s10118" xml:space="preserve">hoc eſt DB. </s>
  <s xml:id="echoid-s10119" xml:space="preserve">DC:</s>
  <s xml:id="echoid-s10120" xml:space="preserve">: DC x DB. </s>
  <s xml:id="echoid-s10121" xml:space="preserve">CN x P. </s>
  <s xml:id="echoid-s10122" xml:space="preserve">hoc <lb/>eſt DB x DC. </s>
  <s xml:id="echoid-s10123" xml:space="preserve">DC q:</s>
  <s xml:id="echoid-s10124" xml:space="preserve">: DC x DB. </s>
  <s xml:id="echoid-s10125" xml:space="preserve">CN x P. </s>
  <s xml:id="echoid-s10126" xml:space="preserve">Quaproptcr eſt <lb/>DC q = CN x P; </s>
  <s xml:id="echoid-s10127" xml:space="preserve">ergò patet _curvam_ DNN _eſſe parabolam_, cu-<lb/>jus _parameter_ P, _vertex_ D; </s>
  <s xml:id="echoid-s10128" xml:space="preserve">_diameter_ ipſi BA parallela.</s>
  <s xml:id="echoid-s10129" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10130" xml:space="preserve">Dedit hoc _Gregorius<unsure/>_ à S. </s>
  <s xml:id="echoid-s10131" xml:space="preserve">_Vincentio_,<anchor type="note" xlink:href="" symbol="*"/> ſed operosâ (ſi probè menmini)
<anchor type="note" xlink:label="note-0229-01a" xlink:href="note-0229-01"/>
prolixitate, demonſtratum.</s>
  <s xml:id="echoid-s10132" xml:space="preserve"/>
</p>
<div xml:id="echoid-div269" type="float" level="2" n="48">
<note symbol="* it" position="right" xlink:label="note-0229-01" xlink:href="note-0229-01a" xml:space="preserve">In Lib. de <lb/>Spirali.</note>
</div>
<p>
  <s xml:id="echoid-s10133" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s10134" xml:space="preserve">Adjicimus; </s>
  <s xml:id="echoid-s10135" xml:space="preserve">Si reliquis iiſdem poſitis, ità ferantur CX, &amp; </s>
  <s xml:id="echoid-s10136" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0229-02a" xlink:href="note-0229-02"/>
DY, ut jam ſemper habeant BE, BC rationem eandem (puta quam <lb/>BD ad R) erunt etiam interſectiones ad _par abolam_.</s>
  <s xml:id="echoid-s10137" xml:space="preserve"/>
</p>
<div xml:id="echoid-div270" type="float" level="2" n="49">
<note position="right" xlink:label="note-0229-02" xlink:href="note-0229-02a" xml:space="preserve">Fig. 48.</note>
</div>
<p>
  <s xml:id="echoid-s10138" xml:space="preserve">Nam biſecetur DB in G, ducatúrque GV ad BE parallela, ſe-<lb/>cans curvam DNN in V; </s>
  <s xml:id="echoid-s10139" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10140" xml:space="preserve">quoniam eſt BC. </s>
  <s xml:id="echoid-s10141" xml:space="preserve">R:</s>
  <s xml:id="echoid-s10142" xml:space="preserve">: BE. </s>
  <s xml:id="echoid-s10143" xml:space="preserve">BD:</s>
  <s xml:id="echoid-s10144" xml:space="preserve">: <lb/>CN. </s>
  <s xml:id="echoid-s10145" xml:space="preserve">CD. </s>
  <s xml:id="echoid-s10146" xml:space="preserve">erit BC x CD = R x CN. </s>
  <s xml:id="echoid-s10147" xml:space="preserve">ergò (ſecundum benè <lb/>notam _parabolæ proprietatem_) eſt curva DNN _parabola_, cujus _para-_ <lb/>_meter_ R, _diameter_ GV.</s>
  <s xml:id="echoid-s10148" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10149" xml:space="preserve">Proletaria ſunt forſan iſta; </s>
  <s xml:id="echoid-s10150" xml:space="preserve">ſed non perinde notata occurunt hæc:</s>
  <s xml:id="echoid-s10151" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10152" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s10153" xml:space="preserve">Si reliquis ſimiliter poſitis, recta CX non jam ad ipſam BA, <lb/>
<anchor type="note" xlink:label="note-0229-03a" xlink:href="note-0229-03"/>
ſed ad aliam poſitione datam (DH) feratur parallela; </s>
  <s xml:id="echoid-s10154" xml:space="preserve">sitque per-<lb/>petuò BE. </s>
  <s xml:id="echoid-s10155" xml:space="preserve">DC:</s>
  <s xml:id="echoid-s10156" xml:space="preserve">: DB. </s>
  <s xml:id="echoid-s10157" xml:space="preserve">R; </s>
  <s xml:id="echoid-s10158" xml:space="preserve">erunt _interſectiones_ N ad _hyperbolam_.</s>
  <s xml:id="echoid-s10159" xml:space="preserve"/>
</p>
<div xml:id="echoid-div271" type="float" level="2" n="50">
<note position="right" xlink:label="note-0229-03" xlink:href="note-0229-03a" xml:space="preserve">Fig. 49.</note>
</div>
<p>
  <s xml:id="echoid-s10160" xml:space="preserve">Nam ductâ NG ad BA parallelâ, nuncupentur DB = _b_. </s>
  <s xml:id="echoid-s10161" xml:space="preserve">BH = <lb/>_h_; </s>
  <s xml:id="echoid-s10162" xml:space="preserve">DG = _x_. </s>
  <s xml:id="echoid-s10163" xml:space="preserve">GN = _y_. </s>
  <s xml:id="echoid-s10164" xml:space="preserve">Eſtque _x.</s>
  <s xml:id="echoid-s10165" xml:space="preserve">y:</s>
  <s xml:id="echoid-s10166" xml:space="preserve">: b_. </s>
  <s xml:id="echoid-s10167" xml:space="preserve">{_by_/_x_} = BE. </s>
  <s xml:id="echoid-s10168" xml:space="preserve">item _h._ <lb/></s>
  <s xml:id="echoid-s10169" xml:space="preserve">_b:</s>
  <s xml:id="echoid-s10170" xml:space="preserve">: y._ </s>
  <s xml:id="echoid-s10171" xml:space="preserve">{_by_/_h_} = GC. </s>
  <s xml:id="echoid-s10172" xml:space="preserve">quare CD = _x_ - {_by_/_h_}. </s>
  <s xml:id="echoid-s10173" xml:space="preserve">Eſt igitur (ex hypotheſi) <lb/>{_by_/_x_}._</s>
  <s xml:id="echoid-s10174" xml:space="preserve">x_ - {_by_/_h_}:</s>
  <s xml:id="echoid-s10175" xml:space="preserve">: _b. </s>
  <s xml:id="echoid-s10176" xml:space="preserve">r_; </s>
  <s xml:id="echoid-s10177" xml:space="preserve">unde talis ordinabitur æquatio; </s>
  <s xml:id="echoid-s10178" xml:space="preserve">_y x_ + {_bry_/_b_} = <lb/>{_h_/_b_}_x x_. </s>
  <s xml:id="echoid-s10179" xml:space="preserve">ponendóq; </s>
  <s xml:id="echoid-s10180" xml:space="preserve">{_hr_/_b_} = _m_; </s>
  <s xml:id="echoid-s10181" xml:space="preserve">erit _yx_ + _my_ = {_m_/_r_}_x x_; </s>
  <s xml:id="echoid-s10182" xml:space="preserve">eſt ergò <lb/>curva DNN _hyperbola_,<anchor type="note" xlink:href="" symbol="*"/> quæ ſuprà habetur determinata.</s>
  <s xml:id="echoid-s10183" xml:space="preserve"/>
</p>
<note symbol="* it" position="right" xml:space="preserve">In 10 hujus.</note>
<p>
  <s xml:id="echoid-s10184" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s10185" xml:space="preserve">Quinetiam ſi (reliquis, ut in præcedente, ſuppoſitis) ità <lb/>jam feratur CX, ut ſemper habeat BE ad BC rationem eandem, quam <lb/>BD ad R; </s>
  <s xml:id="echoid-s10186" xml:space="preserve">erunt itidem interſectiones N ad _hyperbolam_.</s>
  <s xml:id="echoid-s10187" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10188" xml:space="preserve">Nam ductâ NG ad AB parallelâ, nominentur rectæ, ut in præ-<lb/>eunte; </s>
  <s xml:id="echoid-s10189" xml:space="preserve">èſtque jam BC = _b_ - _x_ + {_by_/_b_}; </s>
  <s xml:id="echoid-s10190" xml:space="preserve">atque {_by_/_x_}._</s>
  <s xml:id="echoid-s10191" xml:space="preserve">b_ - _x_ +
<pb o="52" file="0230" n="245" rhead=""/>
{_by_/_b_}:</s>
  <s xml:id="echoid-s10192" xml:space="preserve">: _b. </s>
  <s xml:id="echoid-s10193" xml:space="preserve">r_. </s>
  <s xml:id="echoid-s10194" xml:space="preserve">unde talis emerget æquatio: </s>
  <s xml:id="echoid-s10195" xml:space="preserve">_yx_ + _hx_ - {_hr_/_b_}_y_ = {_h_/_b_} <lb/>_x x x_; </s>
  <s xml:id="echoid-s10196" xml:space="preserve">hoc eſt (poſito {_hr_/_b_} = _m_) _yx_ + _hx_ - _my_ = {_m_/_r_} _xx_; </s>
  <s xml:id="echoid-s10197" xml:space="preserve">Eſt <lb/>
<anchor type="note" xlink:label="note-0230-01a" xlink:href="note-0230-01"/>
igitur curva BNN _hyperbola_, qualem ſuperiùs exhibuimus determi-<lb/>natam.</s>
  <s xml:id="echoid-s10198" xml:space="preserve"/>
</p>
<div xml:id="echoid-div272" type="float" level="2" n="51">
<note position="left" xlink:label="note-0230-01" xlink:href="note-0230-01a" xml:space="preserve">Fig. 50, <lb/>51, <lb/>52.</note>
</div>
<p>
  <s xml:id="echoid-s10199" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s10200" xml:space="preserve">Datæ poſitione ſint rectæ DB, BA; </s>
  <s xml:id="echoid-s10201" xml:space="preserve">(&amp; </s>
  <s xml:id="echoid-s10202" xml:space="preserve">in DB deſigne-<lb/>tur punctum D) ſitque linea DNN talis, ut ductâ utcunque GN <lb/>
<anchor type="note" xlink:label="note-0230-02a" xlink:href="note-0230-02"/>
ad BA parallelâ; </s>
  <s xml:id="echoid-s10203" xml:space="preserve">ſumptis verò determinatis _g, r,_ vocatíſque DG <lb/> = _x_; </s>
  <s xml:id="echoid-s10204" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10205" xml:space="preserve">GN = _y_, ſit _ry_ - _yx_ = _gx_; </s>
  <s xml:id="echoid-s10206" xml:space="preserve">erit linea DNN _by-_ <lb/>_perbola_, ſic determinanda.</s>
  <s xml:id="echoid-s10207" xml:space="preserve"/>
</p>
<div xml:id="echoid-div273" type="float" level="2" n="52">
<note position="right" xlink:label="note-0230-02" xlink:href="note-0230-02a" xml:space="preserve">Fig. 53.</note>
</div>
<p>
  <s xml:id="echoid-s10208" xml:space="preserve">Capiatur DE = _r_, &amp; </s>
  <s xml:id="echoid-s10209" xml:space="preserve">BO = _g_; </s>
  <s xml:id="echoid-s10210" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10211" xml:space="preserve">per E ducatur recta ER ad <lb/>BA, ac per O recta OS ad BD parallelæ; </s>
  <s xml:id="echoid-s10212" xml:space="preserve">erunt ZR, ZS _aſym_-<lb/>_ptoti_.</s>
  <s xml:id="echoid-s10213" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10214" xml:space="preserve">Nam ductâ NP ad DB parallelâ, eſt ZP = _g_ + _y_; </s>
  <s xml:id="echoid-s10215" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10216" xml:space="preserve">PN = <lb/>_r_ - _x_; </s>
  <s xml:id="echoid-s10217" xml:space="preserve">quare ZP x PN = _gr_ - _gx_ + _ry_ - _yx_. </s>
  <s xml:id="echoid-s10218" xml:space="preserve">Verùm ex <lb/>hypotheſi eſt - _gx_ + _ry_ - _yx_ = _o_. </s>
  <s xml:id="echoid-s10219" xml:space="preserve">ergò ZP x PN = _gr_ = <lb/>ZE x ED. </s>
  <s xml:id="echoid-s10220" xml:space="preserve">undè liquido conſtat Propoſitum.</s>
  <s xml:id="echoid-s10221" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10222" xml:space="preserve">Quòd ſi fuerit æquatio _x y - r y = g x_; </s>
  <s xml:id="echoid-s10223" xml:space="preserve">ſumenda eſt DE = _r_; <lb/></s>
  <s xml:id="echoid-s10224" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10225" xml:space="preserve">BO = _g_ (infra rectam DB) ductíſque, ceu priùs, parallelis <lb/>SZR; </s>
  <s xml:id="echoid-s10226" xml:space="preserve">erit _hyperbola_ NNN angulo SZR comprehenſa; </s>
  <s xml:id="echoid-s10227" xml:space="preserve">quod eo-<lb/>dem facilè comprobatur modo.</s>
  <s xml:id="echoid-s10228" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10229" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s10230" xml:space="preserve">Datæ poſitione ſint rectæ DB, BA; </s>
  <s xml:id="echoid-s10231" xml:space="preserve">ac ità ferantur rectæ <lb/>FX ad DB parallela, ac DY per punctum deſignatum D tranſiens, <lb/>ut ſit ſemper ratio ipſius BE ad ipſam BF æqualis aſſignatæ DB ad <lb/>R; </s>
  <s xml:id="echoid-s10232" xml:space="preserve">erunt rectarum DY, FX interſectiones ad lineam rectam.</s>
  <s xml:id="echoid-s10233" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10234" xml:space="preserve">Nam per N ducatur GK ad BA parallela; </s>
  <s xml:id="echoid-s10235" xml:space="preserve">éſtque DB. </s>
  <s xml:id="echoid-s10236" xml:space="preserve">DG:</s>
  <s xml:id="echoid-s10237" xml:space="preserve">: <lb/>BE. </s>
  <s xml:id="echoid-s10238" xml:space="preserve">GN:</s>
  <s xml:id="echoid-s10239" xml:space="preserve">: BE. </s>
  <s xml:id="echoid-s10240" xml:space="preserve">BF:</s>
  <s xml:id="echoid-s10241" xml:space="preserve">: BD. </s>
  <s xml:id="echoid-s10242" xml:space="preserve">R. </s>
  <s xml:id="echoid-s10243" xml:space="preserve">itaque ſemper eſt DG = R. </s>
  <s xml:id="echoid-s10244" xml:space="preserve">Pa-<lb/>tet igitur factâ DG = R, &amp; </s>
  <s xml:id="echoid-s10245" xml:space="preserve">ductâ GK ad BA parallelâ, interſecti-<lb/>ones omnes ad hanc exiſtere.</s>
  <s xml:id="echoid-s10246" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10247" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s10248" xml:space="preserve">Quòd ſi reliquis ſimiliter poſitis; </s>
  <s xml:id="echoid-s10249" xml:space="preserve">ſumpto autem alio in BA <lb/>puncto O; </s>
  <s xml:id="echoid-s10250" xml:space="preserve">ab hoc ſumatur computandi initium; </s>
  <s xml:id="echoid-s10251" xml:space="preserve">ut nimirùm ſit <lb/>perpetuò BE, OF:</s>
  <s xml:id="echoid-s10252" xml:space="preserve">: DB. </s>
  <s xml:id="echoid-s10253" xml:space="preserve">R; </s>
  <s xml:id="echoid-s10254" xml:space="preserve">erunt interſectiones N ad _hyper-_ <lb/>_bolam_.</s>
  <s xml:id="echoid-s10255" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10256" xml:space="preserve">Nam ductâ NG ad AB parallelâ, ſit DB = _b_; </s>
  <s xml:id="echoid-s10257" xml:space="preserve">OB = _g_; </s>
  <s xml:id="echoid-s10258" xml:space="preserve">DG
<pb o="53" file="0231" n="246" rhead=""/>
= x; </s>
  <s xml:id="echoid-s10259" xml:space="preserve">GN = y. </s>
  <s xml:id="echoid-s10260" xml:space="preserve">ergò BE = {by/x}; </s>
  <s xml:id="echoid-s10261" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10262" xml:space="preserve">OF = g + y; </s>
  <s xml:id="echoid-s10263" xml:space="preserve">ergò {by/x}. <lb/></s>
  <s xml:id="echoid-s10264" xml:space="preserve">g + y :</s>
  <s xml:id="echoid-s10265" xml:space="preserve">: b. </s>
  <s xml:id="echoid-s10266" xml:space="preserve">r; </s>
  <s xml:id="echoid-s10267" xml:space="preserve">hinc autem æquatio ry - yx = gx. </s>
  <s xml:id="echoid-s10268" xml:space="preserve">unde DNN <lb/>eſt _hyperbola_ ſuprà mox determinata.</s>
  <s xml:id="echoid-s10269" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10270" xml:space="preserve">Quòd ſi punctum O ſumatur infra D B; </s>
  <s xml:id="echoid-s10271" xml:space="preserve">ſiet æquatio _yx_ - _ry_ = <lb/>_g x_. </s>
  <s xml:id="echoid-s10272" xml:space="preserve">unde rurſus conſtat.</s>
  <s xml:id="echoid-s10273" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10274" xml:space="preserve">XXI. </s>
  <s xml:id="echoid-s10275" xml:space="preserve">Quinetiam, reliquis ſimiliter poſitis, recta FX non jam <lb/>ipſi D B, ſed alteri DH feratur parallela; </s>
  <s xml:id="echoid-s10276" xml:space="preserve">ità ut aſſumpto in B A <lb/>
<anchor type="note" xlink:label="note-0231-01a" xlink:href="note-0231-01"/>
puncto habeat ſemper BE ad OF rationem aſſignatam (DB ad _m_) <lb/>erunt interſectiones N itidem ad _hyperbolam._</s>
  <s xml:id="echoid-s10277" xml:space="preserve"/>
</p>
<div xml:id="echoid-div274" type="float" level="2" n="53">
<note position="right" xlink:label="note-0231-01" xlink:href="note-0231-01a" xml:space="preserve">Fig. 54.</note>
</div>
<p>
  <s xml:id="echoid-s10278" xml:space="preserve">Nam ducatur NG ad AB parallela; </s>
  <s xml:id="echoid-s10279" xml:space="preserve">vocentúrque DB = b; </s>
  <s xml:id="echoid-s10280" xml:space="preserve">HB <lb/>= f; </s>
  <s xml:id="echoid-s10281" xml:space="preserve">HO = g; </s>
  <s xml:id="echoid-s10282" xml:space="preserve">DG = x; </s>
  <s xml:id="echoid-s10283" xml:space="preserve">GN = y; </s>
  <s xml:id="echoid-s10284" xml:space="preserve">eſt ergò x. </s>
  <s xml:id="echoid-s10285" xml:space="preserve">y :</s>
  <s xml:id="echoid-s10286" xml:space="preserve">: b. </s>
  <s xml:id="echoid-s10287" xml:space="preserve">{by/x} <lb/>= BE; </s>
  <s xml:id="echoid-s10288" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10289" xml:space="preserve">b. </s>
  <s xml:id="echoid-s10290" xml:space="preserve">f :</s>
  <s xml:id="echoid-s10291" xml:space="preserve">: x. </s>
  <s xml:id="echoid-s10292" xml:space="preserve">{fx/b} = GK; </s>
  <s xml:id="echoid-s10293" xml:space="preserve">quare NK (FH) = y + {fx/b} <lb/>&amp; </s>
  <s xml:id="echoid-s10294" xml:space="preserve">OF = y + {fx/b} - g. </s>
  <s xml:id="echoid-s10295" xml:space="preserve">Eſt ergò{by/x}. </s>
  <s xml:id="echoid-s10296" xml:space="preserve">y + {fx/b} - g :</s>
  <s xml:id="echoid-s10297" xml:space="preserve">: b. </s>
  <s xml:id="echoid-s10298" xml:space="preserve">m. <lb/></s>
  <s xml:id="echoid-s10299" xml:space="preserve">unde reſultat æquatio my + gx - yx = {f/b}xx. </s>
  <s xml:id="echoid-s10300" xml:space="preserve">vel facto f. </s>
  <s xml:id="echoid-s10301" xml:space="preserve">b :</s>
  <s xml:id="echoid-s10302" xml:space="preserve">: <lb/>m. </s>
  <s xml:id="echoid-s10303" xml:space="preserve">r; </s>
  <s xml:id="echoid-s10304" xml:space="preserve">eſt my + gx - yx = {m/r}x x. </s>
  <s xml:id="echoid-s10305" xml:space="preserve">Conſtat igitur lineam DNN <lb/>eſſe _hyperbolam_; </s>
  <s xml:id="echoid-s10306" xml:space="preserve">qualis ſuperjùs habetur determinata.</s>
  <s xml:id="echoid-s10307" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10308" xml:space="preserve">Notetur, Si computatio ab ipſo puncto H. </s>
  <s xml:id="echoid-s10309" xml:space="preserve">initium ſumat, (hoc eſt <lb/>ſit BE. </s>
  <s xml:id="echoid-s10310" xml:space="preserve">HF :</s>
  <s xml:id="echoid-s10311" xml:space="preserve">: DB. </s>
  <s xml:id="echoid-s10312" xml:space="preserve">m) evaneſcente tunc termino g; </s>
  <s xml:id="echoid-s10313" xml:space="preserve">erit my - yx <lb/>= {m/r}x x; </s>
  <s xml:id="echoid-s10314" xml:space="preserve">unde quoque ſuprà habetur alìa determinatio ſimpli-<lb/>cior.</s>
  <s xml:id="echoid-s10315" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10316" xml:space="preserve">XXII. </s>
  <s xml:id="echoid-s10317" xml:space="preserve">Eſto triangulum ADB, &amp; </s>
  <s xml:id="echoid-s10318" xml:space="preserve">linea DYY talis, ut ductâ ut-<lb/>cunque PM ad DB parallelâ, ſit perpetuò PY = √: </s>
  <s xml:id="echoid-s10319" xml:space="preserve">PMq -<lb/>DBq; </s>
  <s xml:id="echoid-s10320" xml:space="preserve">erit linea DYY _hyperbola_; </s>
  <s xml:id="echoid-s10321" xml:space="preserve">cujus utique Centrum eſt A, ſe-<lb/>_midiameter_ AD, (vel _aſymptotos_ AB) _ſemiparameter_ autem P ; </s>
  <s xml:id="echoid-s10322" xml:space="preserve">faci-<lb/>
<anchor type="note" xlink:label="note-0231-02a" xlink:href="note-0231-02"/>
endo AD. </s>
  <s xml:id="echoid-s10323" xml:space="preserve">DB :</s>
  <s xml:id="echoid-s10324" xml:space="preserve">: DB.</s>
  <s xml:id="echoid-s10325" xml:space="preserve">P.</s>
  <s xml:id="echoid-s10326" xml:space="preserve"/>
</p>
<div xml:id="echoid-div275" type="float" level="2" n="54">
<note position="right" xlink:label="note-0231-02" xlink:href="note-0231-02a" xml:space="preserve">Fig. 55.</note>
</div>
<p>
  <s xml:id="echoid-s10327" xml:space="preserve">Sit enim TD = 2AD. </s>
  <s xml:id="echoid-s10328" xml:space="preserve">Eſtque ADP :</s>
  <s xml:id="echoid-s10329" xml:space="preserve">: (ADq. </s>
  <s xml:id="echoid-s10330" xml:space="preserve">DBq :</s>
  <s xml:id="echoid-s10331" xml:space="preserve">: <lb/>
<anchor type="note" xlink:label="note-0231-03a" xlink:href="note-0231-03"/>
APq. </s>
  <s xml:id="echoid-s10332" xml:space="preserve">PMq :</s>
  <s xml:id="echoid-s10333" xml:space="preserve">: *TP x DP + ADq. </s>
  <s xml:id="echoid-s10334" xml:space="preserve">PMq :</s>
  <s xml:id="echoid-s10335" xml:space="preserve">: TP x DP. <lb/></s>
  <s xml:id="echoid-s10336" xml:space="preserve">PMq - DBq :</s>
  <s xml:id="echoid-s10337" xml:space="preserve">:) TP x DP. </s>
  <s xml:id="echoid-s10338" xml:space="preserve">PYq. </s>
  <s xml:id="echoid-s10339" xml:space="preserve">vel TD. </s>
  <s xml:id="echoid-s10340" xml:space="preserve">2P; </s>
  <s xml:id="echoid-s10341" xml:space="preserve">TP x DP. </s>
  <s xml:id="echoid-s10342" xml:space="preserve"><lb/>PYq. </s>
  <s xml:id="echoid-s10343" xml:space="preserve">unde liquet Propoſitum.</s>
  <s xml:id="echoid-s10344" xml:space="preserve"/>
</p>
<div xml:id="echoid-div276" type="float" level="2" n="55">
<note position="right" xlink:label="note-0231-03" xlink:href="note-0231-03a" xml:space="preserve">* 6, 2. El@@.</note>
</div>
<pb o="54" file="0232" n="247" rhead=""/>
<p>
  <s xml:id="echoid-s10345" xml:space="preserve">_Corol_. </s>
  <s xml:id="echoid-s10346" xml:space="preserve">Si YS tangat _hyperbolam_ DYY; </s>
  <s xml:id="echoid-s10347" xml:space="preserve">erit PMq. </s>
  <s xml:id="echoid-s10348" xml:space="preserve">PYq :</s>
  <s xml:id="echoid-s10349" xml:space="preserve">: <lb/>PA. </s>
  <s xml:id="echoid-s10350" xml:space="preserve">PS.</s>
  <s xml:id="echoid-s10351" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10352" xml:space="preserve">Nam eſt PMq. </s>
  <s xml:id="echoid-s10353" xml:space="preserve">DBq :</s>
  <s xml:id="echoid-s10354" xml:space="preserve">: PAq. </s>
  <s xml:id="echoid-s10355" xml:space="preserve">ADq :</s>
  <s xml:id="echoid-s10356" xml:space="preserve">: PA. </s>
  <s xml:id="echoid-s10357" xml:space="preserve">AS. </s>
  <s xml:id="echoid-s10358" xml:space="preserve">ergò per <lb/>rationis converſionem eſt PMq. </s>
  <s xml:id="echoid-s10359" xml:space="preserve">PYq :</s>
  <s xml:id="echoid-s10360" xml:space="preserve">: PA. </s>
  <s xml:id="echoid-s10361" xml:space="preserve">PS.</s>
  <s xml:id="echoid-s10362" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10363" xml:space="preserve">XXIII. </s>
  <s xml:id="echoid-s10364" xml:space="preserve">Quòd ſi reliquis ſimiliter poſitis; </s>
  <s xml:id="echoid-s10365" xml:space="preserve">ſit jam PY = √ PMq <lb/>
<anchor type="note" xlink:label="note-0232-01a" xlink:href="note-0232-01"/>
+ DBq; </s>
  <s xml:id="echoid-s10366" xml:space="preserve">erit etiam linea YYY _hyperbola_; </s>
  <s xml:id="echoid-s10367" xml:space="preserve">cujus nempe Cen-<lb/>trum A; </s>
  <s xml:id="echoid-s10368" xml:space="preserve">_Semidiameter_ AF (parallela &amp; </s>
  <s xml:id="echoid-s10369" xml:space="preserve">æqualis ipſi DB) _Semi_-<lb/>_parameter_ autem P, ſi ſiat AF. </s>
  <s xml:id="echoid-s10370" xml:space="preserve">AD :</s>
  <s xml:id="echoid-s10371" xml:space="preserve">: AD.</s>
  <s xml:id="echoid-s10372" xml:space="preserve">P.</s>
  <s xml:id="echoid-s10373" xml:space="preserve"/>
</p>
<div xml:id="echoid-div277" type="float" level="2" n="56">
<note position="left" xlink:label="note-0232-01" xlink:href="note-0232-01a" xml:space="preserve">Fig. 56.</note>
</div>
<p>
  <s xml:id="echoid-s10374" xml:space="preserve">Nam ducatur YK ipſi AP parallela cum AF conveniens in K; <lb/></s>
  <s xml:id="echoid-s10375" xml:space="preserve">Sítque FT = 2 FA; </s>
  <s xml:id="echoid-s10376" xml:space="preserve">éſtque AF. </s>
  <s xml:id="echoid-s10377" xml:space="preserve">P :</s>
  <s xml:id="echoid-s10378" xml:space="preserve">: (AFq. </s>
  <s xml:id="echoid-s10379" xml:space="preserve">ADq :</s>
  <s xml:id="echoid-s10380" xml:space="preserve">: DBq. </s>
  <s xml:id="echoid-s10381" xml:space="preserve"><lb/>ADq :</s>
  <s xml:id="echoid-s10382" xml:space="preserve">: PMq. </s>
  <s xml:id="echoid-s10383" xml:space="preserve">APq :</s>
  <s xml:id="echoid-s10384" xml:space="preserve">: PYq - DBq. </s>
  <s xml:id="echoid-s10385" xml:space="preserve">APq :</s>
  <s xml:id="echoid-s10386" xml:space="preserve">: AKq - AFq. </s>
  <s xml:id="echoid-s10387" xml:space="preserve"><lb/>KYq :</s>
  <s xml:id="echoid-s10388" xml:space="preserve">:) TK x FK. </s>
  <s xml:id="echoid-s10389" xml:space="preserve">KYq :</s>
  <s xml:id="echoid-s10390" xml:space="preserve">: AF.</s>
  <s xml:id="echoid-s10391" xml:space="preserve">P. </s>
  <s xml:id="echoid-s10392" xml:space="preserve">unde conſtat Propoſi-<lb/>tum.</s>
  <s xml:id="echoid-s10393" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10394" xml:space="preserve">_Corol_. </s>
  <s xml:id="echoid-s10395" xml:space="preserve">Rurſus, Si recta YS _hyperbolam_ FYY tangat, erit PMq. <lb/></s>
  <s xml:id="echoid-s10396" xml:space="preserve">PYq :</s>
  <s xml:id="echoid-s10397" xml:space="preserve">: PA. </s>
  <s xml:id="echoid-s10398" xml:space="preserve">PS.</s>
  <s xml:id="echoid-s10399" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10400" xml:space="preserve">Nam AD eſt _Semidiameter_ ipſi AF conjugata. </s>
  <s xml:id="echoid-s10401" xml:space="preserve">unde PA. </s>
  <s xml:id="echoid-s10402" xml:space="preserve">AS :</s>
  <s xml:id="echoid-s10403" xml:space="preserve">: <lb/>PAq. </s>
  <s xml:id="echoid-s10404" xml:space="preserve">ADq :</s>
  <s xml:id="echoid-s10405" xml:space="preserve">: PMq. </s>
  <s xml:id="echoid-s10406" xml:space="preserve">DBq. </s>
  <s xml:id="echoid-s10407" xml:space="preserve">ergò PA. </s>
  <s xml:id="echoid-s10408" xml:space="preserve">PS :</s>
  <s xml:id="echoid-s10409" xml:space="preserve">: PMq. </s>
  <s xml:id="echoid-s10410" xml:space="preserve">PMq <lb/>+ DBq :</s>
  <s xml:id="echoid-s10411" xml:space="preserve">: PMq. </s>
  <s xml:id="echoid-s10412" xml:space="preserve">PYq.</s>
  <s xml:id="echoid-s10413" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10414" xml:space="preserve">XXIV. </s>
  <s xml:id="echoid-s10415" xml:space="preserve">Sit triangulum ADB, rectum habens angulum ADB; <lb/></s>
  <s xml:id="echoid-s10416" xml:space="preserve">
<anchor type="note" xlink:label="note-0232-02a" xlink:href="note-0232-02"/>
&amp; </s>
  <s xml:id="echoid-s10417" xml:space="preserve">curva CGD talis, ut ductâ quâcunque rectâ FEG ad DB paral-<lb/>lelâ (quæ lineas expoſitas ſecet ut vides) ſit aggregatum quadrato-<lb/>rum ex EF, EG æquale quadrato ex DB; </s>
  <s xml:id="echoid-s10418" xml:space="preserve">erit curva CGD _εllip_-<lb/>_ſis_ cujus ſemiaxes AD, AC.</s>
  <s xml:id="echoid-s10419" xml:space="preserve"/>
</p>
<div xml:id="echoid-div278" type="float" level="2" n="57">
<note position="left" xlink:label="note-0232-02" xlink:href="note-0232-02a" xml:space="preserve">Fig. 57.</note>
</div>
<p>
  <s xml:id="echoid-s10420" xml:space="preserve">Nam ſit AV = AD. </s>
  <s xml:id="echoid-s10421" xml:space="preserve">Eſtque ADq. </s>
  <s xml:id="echoid-s10422" xml:space="preserve">DBq (ACq) :</s>
  <s xml:id="echoid-s10423" xml:space="preserve">: AEq. <lb/></s>
  <s xml:id="echoid-s10424" xml:space="preserve">EFq :</s>
  <s xml:id="echoid-s10425" xml:space="preserve">: ADq - AEq. </s>
  <s xml:id="echoid-s10426" xml:space="preserve">DBq - EFq. </s>
  <s xml:id="echoid-s10427" xml:space="preserve">Hoc eſt ADq. </s>
  <s xml:id="echoid-s10428" xml:space="preserve">ACq :</s>
  <s xml:id="echoid-s10429" xml:space="preserve">: <lb/>VE x ED. </s>
  <s xml:id="echoid-s10430" xml:space="preserve">EGq. </s>
  <s xml:id="echoid-s10431" xml:space="preserve">unde liquet Propoſitum.</s>
  <s xml:id="echoid-s10432" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10433" xml:space="preserve">_Nota_, Tangat GT _ellipſin_ CGD; </s>
  <s xml:id="echoid-s10434" xml:space="preserve">eſt EFq. </s>
  <s xml:id="echoid-s10435" xml:space="preserve">EGq :</s>
  <s xml:id="echoid-s10436" xml:space="preserve">: EA. <lb/></s>
  <s xml:id="echoid-s10437" xml:space="preserve">ET.</s>
  <s xml:id="echoid-s10438" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10439" xml:space="preserve">Nam ob AE. </s>
  <s xml:id="echoid-s10440" xml:space="preserve">AD :</s>
  <s xml:id="echoid-s10441" xml:space="preserve">: AD. </s>
  <s xml:id="echoid-s10442" xml:space="preserve">AT. </s>
  <s xml:id="echoid-s10443" xml:space="preserve">eſt AEq. </s>
  <s xml:id="echoid-s10444" xml:space="preserve">ADq :</s>
  <s xml:id="echoid-s10445" xml:space="preserve">: AE. </s>
  <s xml:id="echoid-s10446" xml:space="preserve">AT. <lb/></s>
  <s xml:id="echoid-s10447" xml:space="preserve">unde AEq. </s>
  <s xml:id="echoid-s10448" xml:space="preserve">ADq - AEq :</s>
  <s xml:id="echoid-s10449" xml:space="preserve">: AE. </s>
  <s xml:id="echoid-s10450" xml:space="preserve">AT - AE. </s>
  <s xml:id="echoid-s10451" xml:space="preserve">Hoc eſt EFq. </s>
  <s xml:id="echoid-s10452" xml:space="preserve"><lb/>DBq - EFq :</s>
  <s xml:id="echoid-s10453" xml:space="preserve">: AE. </s>
  <s xml:id="echoid-s10454" xml:space="preserve">ET. </s>
  <s xml:id="echoid-s10455" xml:space="preserve">hoc eſt EFq. </s>
  <s xml:id="echoid-s10456" xml:space="preserve">EGq :</s>
  <s xml:id="echoid-s10457" xml:space="preserve">: AE. </s>
  <s xml:id="echoid-s10458" xml:space="preserve">ET.</s>
  <s xml:id="echoid-s10459" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10460" xml:space="preserve">Sit _Angulus rectilineus_ DTH, in cujus latere TD ſignetur pun-<lb/>ctum A. </s>
  <s xml:id="echoid-s10461" xml:space="preserve">Sit item curva VGG proprietate talis, ut ductâ rectâ quâ-<lb/>
<anchor type="note" xlink:label="note-0232-03a" xlink:href="note-0232-03"/>
piam EFG ad TD perpendiculari (quæ lineas TD, TH, VGG <lb/>ſecet punctis E, F, G,) connexáque rectâ AF, ſit EG = AF; <lb/></s>
  <s xml:id="echoid-s10462" xml:space="preserve">erit linea VGG _hyperbola_.</s>
  <s xml:id="echoid-s10463" xml:space="preserve"/>
</p>
<div xml:id="echoid-div279" type="float" level="2" n="58">
<note position="left" xlink:label="note-0232-03" xlink:href="note-0232-03a" xml:space="preserve">Fig. 58.</note>
</div>
<p>
  <s xml:id="echoid-s10464" xml:space="preserve">Nam ducantur AP ad TH &amp; </s>
  <s xml:id="echoid-s10465" xml:space="preserve">VPC ad TD perpendiculares;</s>
  <s xml:id="echoid-s10466" xml:space="preserve">
<pb o="55" file="0233" n="248" rhead=""/>
item PO ad TE parallela. </s>
  <s xml:id="echoid-s10467" xml:space="preserve">Eſtque EFq = EOq (CPq) + OFq <lb/>+ 2 EO x OF (+ 2 CP x OF). </s>
  <s xml:id="echoid-s10468" xml:space="preserve">Verùm ob CP. </s>
  <s xml:id="echoid-s10469" xml:space="preserve">CA :</s>
  <s xml:id="echoid-s10470" xml:space="preserve">: OP. <lb/></s>
  <s xml:id="echoid-s10471" xml:space="preserve">OF :</s>
  <s xml:id="echoid-s10472" xml:space="preserve">: CE. </s>
  <s xml:id="echoid-s10473" xml:space="preserve">OF; </s>
  <s xml:id="echoid-s10474" xml:space="preserve">eſt CP x OF = CA x CE; </s>
  <s xml:id="echoid-s10475" xml:space="preserve">ergò EFq = <lb/>CPq + OFq + 2 CA x CE. </s>
  <s xml:id="echoid-s10476" xml:space="preserve">item eſt AEq = CEq + <lb/>CAq - 2 CA x CE; </s>
  <s xml:id="echoid-s10477" xml:space="preserve">quapropter eſt EFq + AEq = CPq + <lb/>CAq + OFq + CEq. </s>
  <s xml:id="echoid-s10478" xml:space="preserve">hoc eſt EGq = (APq + PFq = ) <lb/>CVq + PFq. </s>
  <s xml:id="echoid-s10479" xml:space="preserve">vel EGq - PFq = CVq. </s>
  <s xml:id="echoid-s10480" xml:space="preserve">Verùm eſt CE. </s>
  <s xml:id="echoid-s10481" xml:space="preserve"><lb/>(PO). </s>
  <s xml:id="echoid-s10482" xml:space="preserve">PF :</s>
  <s xml:id="echoid-s10483" xml:space="preserve">: CP.</s>
  <s xml:id="echoid-s10484" xml:space="preserve">AP :</s>
  <s xml:id="echoid-s10485" xml:space="preserve">: CP.</s>
  <s xml:id="echoid-s10486" xml:space="preserve">CV; </s>
  <s xml:id="echoid-s10487" xml:space="preserve">unde EGq - {CVq/CPq} CEq <lb/>= CVq; </s>
  <s xml:id="echoid-s10488" xml:space="preserve">adeóque linea GVG eſt _hyperbola_, cujus centrum C; </s>
  <s xml:id="echoid-s10489" xml:space="preserve">ſe-<lb/>miaxes CV, CP.</s>
  <s xml:id="echoid-s10490" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10491" xml:space="preserve">Not. </s>
  <s xml:id="echoid-s10492" xml:space="preserve">Ductà rectâ FQ ad TH perpendiculari, ſumptáque QR = <lb/>AE; </s>
  <s xml:id="echoid-s10493" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10494" xml:space="preserve">connexâ GR; </s>
  <s xml:id="echoid-s10495" xml:space="preserve">erit GR _hyperbolæ_ VGG perpendicularis; <lb/></s>
  <s xml:id="echoid-s10496" xml:space="preserve">mihi præſta sîs ſidem; </s>
  <s xml:id="echoid-s10497" xml:space="preserve">aut ipſe rem ad Calculum exige; </s>
  <s xml:id="echoid-s10498" xml:space="preserve">eò verba <lb/>non proſundam.</s>
  <s xml:id="echoid-s10499" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10500" xml:space="preserve">XXVI. </s>
  <s xml:id="echoid-s10501" xml:space="preserve">Poſitione datæ ſint rectæ AC, BD (ſe interſecantes in X) <lb/>
<anchor type="note" xlink:label="note-0233-01a" xlink:href="note-0233-01"/>
qnas decuſſet recta A B; </s>
  <s xml:id="echoid-s10502" xml:space="preserve">tum ductâ utcunque rectâ PKL ad AB <lb/>parallelâ, (quæ rectas AC, BD ſecet punctis P, K) ſit PL æqua-<lb/>lis ipſi BK; </s>
  <s xml:id="echoid-s10503" xml:space="preserve">erit linea ALL recta.</s>
  <s xml:id="echoid-s10504" xml:space="preserve"/>
</p>
<div xml:id="echoid-div280" type="float" level="2" n="59">
<note position="right" xlink:label="note-0233-01" xlink:href="note-0233-01a" xml:space="preserve">Fig. 59.</note>
</div>
<p>
  <s xml:id="echoid-s10505" xml:space="preserve">Nam, (ductâ XQ ad BA parallelâ, eſt AQ. </s>
  <s xml:id="echoid-s10506" xml:space="preserve">AP :</s>
  <s xml:id="echoid-s10507" xml:space="preserve">: (BX. <lb/></s>
  <s xml:id="echoid-s10508" xml:space="preserve">BK :</s>
  <s xml:id="echoid-s10509" xml:space="preserve">: ) QX. </s>
  <s xml:id="echoid-s10510" xml:space="preserve">PL: </s>
  <s xml:id="echoid-s10511" xml:space="preserve">ergo linea ALL eſt recta.</s>
  <s xml:id="echoid-s10512" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10513" xml:space="preserve">XXVII. </s>
  <s xml:id="echoid-s10514" xml:space="preserve">Poſitione data ſit recta A X, &amp; </s>
  <s xml:id="echoid-s10515" xml:space="preserve">punctum D; </s>
  <s xml:id="echoid-s10516" xml:space="preserve">neque non <lb/>linea DNN talis; </s>
  <s xml:id="echoid-s10517" xml:space="preserve">ut per D ductâ quâcunque rectâ MN (quæ re-<lb/>
<anchor type="note" xlink:label="note-0233-02a" xlink:href="note-0233-02"/>
ctam AX ſecet in M, &amp; </s>
  <s xml:id="echoid-s10518" xml:space="preserve">lineam DNN in N) ſit perpetim rectangu-<lb/>lum ex DM, DN æquale dato (puta quadrato ex Z); </s>
  <s xml:id="echoid-s10519" xml:space="preserve">erit linea <lb/>DNN circularis.</s>
  <s xml:id="echoid-s10520" xml:space="preserve"/>
</p>
<div xml:id="echoid-div281" type="float" level="2" n="60">
<note position="right" xlink:label="note-0233-02" xlink:href="note-0233-02a" xml:space="preserve">Fig. 60.</note>
</div>
<p>
  <s xml:id="echoid-s10521" xml:space="preserve">Nam ducatur DB ad AX perpendicularis; </s>
  <s xml:id="echoid-s10522" xml:space="preserve">ſitque DB. </s>
  <s xml:id="echoid-s10523" xml:space="preserve">Z :</s>
  <s xml:id="echoid-s10524" xml:space="preserve">: Z. <lb/></s>
  <s xml:id="echoid-s10525" xml:space="preserve">DE; </s>
  <s xml:id="echoid-s10526" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10527" xml:space="preserve">connectatur NE; </s>
  <s xml:id="echoid-s10528" xml:space="preserve">Eſt jam DM x DN = Zq = DB x <lb/>DE; </s>
  <s xml:id="echoid-s10529" xml:space="preserve">quare DM. </s>
  <s xml:id="echoid-s10530" xml:space="preserve">DB :</s>
  <s xml:id="echoid-s10531" xml:space="preserve">: DE. </s>
  <s xml:id="echoid-s10532" xml:space="preserve">DN. </s>
  <s xml:id="echoid-s10533" xml:space="preserve">ergò triangula DBM, DNE <lb/>ſimilia ſunt; </s>
  <s xml:id="echoid-s10534" xml:space="preserve">quapropter angulus DNE rectus eſt; </s>
  <s xml:id="echoid-s10535" xml:space="preserve">itaque linea DNN <lb/>eſt circularis: </s>
  <s xml:id="echoid-s10536" xml:space="preserve">(ad circulum pertinens, _c@jus Diameter_ DE).</s>
  <s xml:id="echoid-s10537" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10538" xml:space="preserve">Vides nedum rectam &amp; </s>
  <s xml:id="echoid-s10539" xml:space="preserve">_hyperbolam_; </s>
  <s xml:id="echoid-s10540" xml:space="preserve">ſed &amp; </s>
  <s xml:id="echoid-s10541" xml:space="preserve">ſuo modo rectam ac <lb/>_circulum_ ſibi lineas eſſe reciprocas. </s>
  <s xml:id="echoid-s10542" xml:space="preserve">Verùm hîc, etſi præludiis no-<lb/>ſtris nondum abſolutis, paulùm ſubſiſtamus.</s>
  <s xml:id="echoid-s10543" xml:space="preserve"/>
</p>
<pb o="56" file="0234" n="249"/>
</div>
<div xml:id="echoid-div283" type="section" level="1" n="32">
<head xml:id="echoid-head35" xml:space="preserve"><emph style="sc">Lect</emph>. VII.</head>
<p>
  <s xml:id="echoid-s10544" xml:space="preserve">ADhuc in _Veſtibulo_ hæremus; </s>
  <s xml:id="echoid-s10545" xml:space="preserve">nec aliud quàm velitamus.</s>
  <s xml:id="echoid-s10546" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10547" xml:space="preserve">I. </s>
  <s xml:id="echoid-s10548" xml:space="preserve">Sint duo quanta A, B; </s>
  <s xml:id="echoid-s10549" xml:space="preserve">quorum majus A; </s>
  <s xml:id="echoid-s10550" xml:space="preserve">adſumpto tertio quo-<lb/>piam X, erit A + X. </s>
  <s xml:id="echoid-s10551" xml:space="preserve">B + X &amp;</s>
  <s xml:id="echoid-s10552" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s10553" xml:space="preserve">A. </s>
  <s xml:id="echoid-s10554" xml:space="preserve">B.</s>
  <s xml:id="echoid-s10555" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10556" xml:space="preserve">Nam ob X. </s>
  <s xml:id="echoid-s10557" xml:space="preserve">A &amp;</s>
  <s xml:id="echoid-s10558" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s10559" xml:space="preserve">X. </s>
  <s xml:id="echoid-s10560" xml:space="preserve">B; </s>
  <s xml:id="echoid-s10561" xml:space="preserve">erit componendo X + A. </s>
  <s xml:id="echoid-s10562" xml:space="preserve">A. </s>
  <s xml:id="echoid-s10563" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s10564" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s10565" xml:space="preserve">X + <lb/>B. </s>
  <s xml:id="echoid-s10566" xml:space="preserve">B. </s>
  <s xml:id="echoid-s10567" xml:space="preserve">vel permutando X + A. </s>
  <s xml:id="echoid-s10568" xml:space="preserve">X + B &amp;</s>
  <s xml:id="echoid-s10569" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s10570" xml:space="preserve">A. </s>
  <s xml:id="echoid-s10571" xml:space="preserve">B.</s>
  <s xml:id="echoid-s10572" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10573" xml:space="preserve">II. </s>
  <s xml:id="echoid-s10574" xml:space="preserve">In linea YZ ſignentur tria puncta, L, M, N; </s>
  <s xml:id="echoid-s10575" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10576" xml:space="preserve">inter puncta <lb/>L, N ſumpto puncto quopiam E, alteróque G extra LN (verſus Z); <lb/></s>
  <s xml:id="echoid-s10577" xml:space="preserve">ſecetur EG in F, ut ſit GE. </s>
  <s xml:id="echoid-s10578" xml:space="preserve">EF :</s>
  <s xml:id="echoid-s10579" xml:space="preserve">: NL. </s>
  <s xml:id="echoid-s10580" xml:space="preserve">LM; </s>
  <s xml:id="echoid-s10581" xml:space="preserve">cadet punctum F ad <lb/>
<anchor type="note" xlink:label="note-0234-01a" xlink:href="note-0234-01"/>
partes MZ:</s>
  <s xml:id="echoid-s10582" xml:space="preserve"/>
</p>
<div xml:id="echoid-div283" type="float" level="2" n="1">
<note position="left" xlink:label="note-0234-01" xlink:href="note-0234-01a" xml:space="preserve">Fig. 61.</note>
</div>
<p>
  <s xml:id="echoid-s10583" xml:space="preserve">Nam eſt N E. </s>
  <s xml:id="echoid-s10584" xml:space="preserve">ME <anchor type="note" xlink:href="" symbol="*"/> &amp;</s>
  <s xml:id="echoid-s10585" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10586" xml:space="preserve">NL. </s>
  <s xml:id="echoid-s10587" xml:space="preserve">ML :</s>
  <s xml:id="echoid-s10588" xml:space="preserve">: GE. </s>
  <s xml:id="echoid-s10589" xml:space="preserve">FE &amp;</s>
  <s xml:id="echoid-s10590" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10591" xml:space="preserve">NE. </s>
  <s xml:id="echoid-s10592" xml:space="preserve">PE.</s>
  <s xml:id="echoid-s10593" xml:space="preserve">
<anchor type="note" xlink:label="note-0234-02a" xlink:href="note-0234-02"/>
ergo FE &amp;</s>
  <s xml:id="echoid-s10594" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10595" xml:space="preserve">ME.</s>
  <s xml:id="echoid-s10596" xml:space="preserve"/>
</p>
<div xml:id="echoid-div284" type="float" level="2" n="2">
<note position="left" xlink:label="note-0234-02" xlink:href="note-0234-02a" xml:space="preserve">* I _hujus_.</note>
</div>
<p>
  <s xml:id="echoid-s10597" xml:space="preserve">III. </s>
  <s xml:id="echoid-s10598" xml:space="preserve">Sint rectæ BA, DC parallelæ item rectæ BD, GP parallelæ; <lb/></s>
  <s xml:id="echoid-s10599" xml:space="preserve">pérque punctum B ducantur utcunque duæ rectæ BT, BS ipſam GP <lb/>
<anchor type="note" xlink:label="note-0234-03a" xlink:href="note-0234-03"/>
ſecantes punctis L, K, dico fore D S. </s>
  <s xml:id="echoid-s10600" xml:space="preserve">DT :</s>
  <s xml:id="echoid-s10601" xml:space="preserve">: KG. </s>
  <s xml:id="echoid-s10602" xml:space="preserve">LG.</s>
  <s xml:id="echoid-s10603" xml:space="preserve"/>
</p>
<div xml:id="echoid-div285" type="float" level="2" n="3">
<note position="left" xlink:label="note-0234-03" xlink:href="note-0234-03a" xml:space="preserve">Fig. 62.</note>
</div>
<p>
  <s xml:id="echoid-s10604" xml:space="preserve">Nam eſt KG. </s>
  <s xml:id="echoid-s10605" xml:space="preserve">LG = KG. </s>
  <s xml:id="echoid-s10606" xml:space="preserve">GB + GB. </s>
  <s xml:id="echoid-s10607" xml:space="preserve">LG = PK. </s>
  <s xml:id="echoid-s10608" xml:space="preserve">PS + <lb/>PT. </s>
  <s xml:id="echoid-s10609" xml:space="preserve">PL = DB. </s>
  <s xml:id="echoid-s10610" xml:space="preserve">DS + DT. </s>
  <s xml:id="echoid-s10611" xml:space="preserve">DB = DT. </s>
  <s xml:id="echoid-s10612" xml:space="preserve">DS.</s>
  <s xml:id="echoid-s10613" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10614" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s10615" xml:space="preserve">Eſto triangulum BDT, basíque DB parallelam quamvis PG <lb/>interſecent per B ductæ quæpiam duæ rectæ BS, BR punctis L, K; <lb/></s>
  <s xml:id="echoid-s10616" xml:space="preserve">dico fore LG x TD + KL x RD. </s>
  <s xml:id="echoid-s10617" xml:space="preserve">KG x TD :</s>
  <s xml:id="echoid-s10618" xml:space="preserve">: RD. </s>
  <s xml:id="echoid-s10619" xml:space="preserve">SD.</s>
  <s xml:id="echoid-s10620" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10621" xml:space="preserve">Sumantur enim BM = GP, &amp; </s>
  <s xml:id="echoid-s10622" xml:space="preserve">BN = LP; </s>
  <s xml:id="echoid-s10623" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10624" xml:space="preserve">BO = KP; <lb/></s>
  <s xml:id="echoid-s10625" xml:space="preserve">
<anchor type="note" xlink:label="note-0234-04a" xlink:href="note-0234-04"/>
unde conſtat junctas PM, PN, PO ipſis TB, SB, RB (reſpectivè) <lb/>parallelas eſſe. </s>
  <s xml:id="echoid-s10626" xml:space="preserve">Et quoniam eſt DM. </s>
  <s xml:id="echoid-s10627" xml:space="preserve">PD :</s>
  <s xml:id="echoid-s10628" xml:space="preserve">: DB. </s>
  <s xml:id="echoid-s10629" xml:space="preserve">TD. </s>
  <s xml:id="echoid-s10630" xml:space="preserve">erit DM x <lb/>TD = PD x <emph style="sc">Db</emph>. </s>
  <s xml:id="echoid-s10631" xml:space="preserve">Similiter eſt DN x SD = PD x DB. </s>
  <s xml:id="echoid-s10632" xml:space="preserve">quare <lb/>DM x TD = DN x SD = DM x SD + MN x SD, tranſpo-<lb/>nendóque DM x TD - DM x SD = MN x SD. </s>
  <s xml:id="echoid-s10633" xml:space="preserve">Simili plané
<pb o="57" file="0235" n="250" rhead=""/>
diſcurſu eſt DM x TD - DM x RD = MO x RD. </s>
  <s xml:id="echoid-s10634" xml:space="preserve">quapropter <lb/>
<anchor type="note" xlink:label="note-0235-01a" xlink:href="note-0235-01"/>
erit MN x SD. </s>
  <s xml:id="echoid-s10635" xml:space="preserve">MO x RD :</s>
  <s xml:id="echoid-s10636" xml:space="preserve">: TD - SD. </s>
  <s xml:id="echoid-s10637" xml:space="preserve">TD - RD. </s>
  <s xml:id="echoid-s10638" xml:space="preserve">hoc eſt <lb/>LG x SD. </s>
  <s xml:id="echoid-s10639" xml:space="preserve">KG x RD :</s>
  <s xml:id="echoid-s10640" xml:space="preserve">: TD - SD. </s>
  <s xml:id="echoid-s10641" xml:space="preserve">TD - RD; </s>
  <s xml:id="echoid-s10642" xml:space="preserve">vel (ad æqua-<lb/>tionem redigendo) LG x SD x TD - LG x SD x RD = KG x <lb/>RD x TD - KG x RD x SD; </s>
  <s xml:id="echoid-s10643" xml:space="preserve">tranſponendóque LG x SD x <lb/>TD + KG x RD x SD - LG x SD x RD = KG x RD x TD. <lb/></s>
  <s xml:id="echoid-s10644" xml:space="preserve">hoc eſt LG x SD x TD + KL x SD x RD = KG x RD x TD. </s>
  <s xml:id="echoid-s10645" xml:space="preserve"><lb/>vel (ad analogiſmum reducendo) LG x TD + KL x RD. </s>
  <s xml:id="echoid-s10646" xml:space="preserve">KG x <lb/>TD :</s>
  <s xml:id="echoid-s10647" xml:space="preserve">: RD. </s>
  <s xml:id="echoid-s10648" xml:space="preserve">SD. </s>
  <s xml:id="echoid-s10649" xml:space="preserve">Quod erat Propoſitum.</s>
  <s xml:id="echoid-s10650" xml:space="preserve"/>
</p>
<div xml:id="echoid-div286" type="float" level="2" n="4">
<note position="left" xlink:label="note-0234-04" xlink:href="note-0234-04a" xml:space="preserve">Fig. 63</note>
<note position="right" xlink:label="note-0235-01" xlink:href="note-0235-01a" xml:space="preserve">Fig. 63.</note>
</div>
<p>
  <s xml:id="echoid-s10651" xml:space="preserve">V. </s>
  <s xml:id="echoid-s10652" xml:space="preserve">Quòd ſi puncta T, R non ad eaſdem puncti D partes ſita ſint, <lb/>
<anchor type="note" xlink:label="note-0235-02a" xlink:href="note-0235-02"/>
erit LG x RD - KL x TD. </s>
  <s xml:id="echoid-s10653" xml:space="preserve">KG x TD :</s>
  <s xml:id="echoid-s10654" xml:space="preserve">: RD. </s>
  <s xml:id="echoid-s10655" xml:space="preserve">SD.</s>
  <s xml:id="echoid-s10656" xml:space="preserve"/>
</p>
<div xml:id="echoid-div287" type="float" level="2" n="5">
<note position="right" xlink:label="note-0235-02" xlink:href="note-0235-02a" xml:space="preserve">Fig. 64.</note>
</div>
<p>
  <s xml:id="echoid-s10657" xml:space="preserve">Simili conſtabit id diſcurſu; </s>
  <s xml:id="echoid-s10658" xml:space="preserve">quem piget repetere.</s>
  <s xml:id="echoid-s10659" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10660" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s10661" xml:space="preserve">Sint quatuor continuè proportionalium ſeries æquinumeræ (qua-<lb/>les adſcriptas cernis) quarum cùm antecedentes primi, tum ultimi conſe-<lb/>quentes inter ſe proportionales ſint(A. </s>
  <s xml:id="echoid-s10662" xml:space="preserve">α:</s>
  <s xml:id="echoid-s10663" xml:space="preserve">: M. </s>
  <s xml:id="echoid-s10664" xml:space="preserve">μ; </s>
  <s xml:id="echoid-s10665" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10666" xml:space="preserve">F. </s>
  <s xml:id="echoid-s10667" xml:space="preserve">φ:</s>
  <s xml:id="echoid-s10668" xml:space="preserve">: S. </s>
  <s xml:id="echoid-s10669" xml:space="preserve">σ) <lb/>crunt ejuſdem ordinis quilibet accepti quatuor etiam inter ſe proportio-<lb/>nales (puta nempe, D. </s>
  <s xml:id="echoid-s10670" xml:space="preserve">δ:</s>
  <s xml:id="echoid-s10671" xml:space="preserve">: P. </s>
  <s xml:id="echoid-s10672" xml:space="preserve">π).</s>
  <s xml:id="echoid-s10673" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve"> <lb/>A. # B. # C. # D. # E. # F. <lb/>α. # β. # γ. # δ. # @. # φ <lb/>M. # N. # O. # P. # R. # S. <lb/>μ. # ν. # ο. # π. # ς. # σ. <lb/></note>
<p>
  <s xml:id="echoid-s10674" xml:space="preserve">Sunt enim Aμ, Bγ, Cο, Dπ, Eς, Fσ, \\ &amp; </s>
  <s xml:id="echoid-s10675" xml:space="preserve">αM, βN, γO, δP, @R, φ S,} Continuè propor-<lb/>tionales.</s>
  <s xml:id="echoid-s10676" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10677" xml:space="preserve">Cùm igitur ſit Aμ, = αM; </s>
  <s xml:id="echoid-s10678" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10679" xml:space="preserve">Fσ = φ S, liquidum eſt ſore D π <lb/>= σ P; </s>
  <s xml:id="echoid-s10680" xml:space="preserve">ac idcircò D. </s>
  <s xml:id="echoid-s10681" xml:space="preserve">δ:</s>
  <s xml:id="echoid-s10682" xml:space="preserve">: P. </s>
  <s xml:id="echoid-s10683" xml:space="preserve">π. </s>
  <s xml:id="echoid-s10684" xml:space="preserve">Ad utramque proportionalitatem <lb/>(tam Arithmeticam quàm Geometricam) æquè ſpectat hæc Con-<lb/>cluſio.</s>
  <s xml:id="echoid-s10685" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10686" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s10687" xml:space="preserve">Rectæ A B, CD parallelæ ſint; </s>
  <s xml:id="echoid-s10688" xml:space="preserve">hásque ſecet poſitione data <lb/>
<anchor type="note" xlink:label="note-0235-04a" xlink:href="note-0235-04"/>
BD; </s>
  <s xml:id="echoid-s10689" xml:space="preserve">lineæ verò EBE; </s>
  <s xml:id="echoid-s10690" xml:space="preserve">FBF ita relatæ ſint, ut ductâ utcunque <lb/>recta PG ad DB parallelâ; </s>
  <s xml:id="echoid-s10691" xml:space="preserve">ſit ſemper PF eodem ordine media pro-<lb/>portionalis inter PG, PE; </s>
  <s xml:id="echoid-s10692" xml:space="preserve">tum per quodvis deſignatum lineæ EBE <lb/>punctum E tranſeat HE ipſis AB, CD parallela, sítque alia curva <lb/>KEK talis, ut ductâ utcunque QL itidem ad DB parallelâ, ſit QX
<pb o="58" file="0236" n="251" rhead=""/>
eodem ſemper ordine media inter QL, QI (eodem inquam illo, quo <lb/>PF media fuerat inter PG, PE) : </s>
  <s xml:id="echoid-s10693" xml:space="preserve">dico lineas FBF, KEK analo-<lb/>
<anchor type="note" xlink:label="note-0236-01a" xlink:href="note-0236-01"/>
gas eſſe; </s>
  <s xml:id="echoid-s10694" xml:space="preserve">hoc eſt ordinatas (quales QR, QK) eandem perpetuò in-<lb/>ter ſe rationem habere; </s>
  <s xml:id="echoid-s10695" xml:space="preserve">eandem ſcilicet illi quam habet PF ad PE.</s>
  <s xml:id="echoid-s10696" xml:space="preserve"/>
</p>
<div xml:id="echoid-div288" type="float" level="2" n="6">
<note position="right" xlink:label="note-0235-04" xlink:href="note-0235-04a" xml:space="preserve">Fig. 65.</note>
<note position="left" xlink:label="note-0236-01" xlink:href="note-0236-01a" xml:space="preserve">Fig. 65.</note>
</div>
<p>
  <s xml:id="echoid-s10697" xml:space="preserve">Hoc è Lemmate proximè præmiſſo conſectatur, utì patebit, ad ſub-<lb/>jectum Schema mentem advertendo.</s>
  <s xml:id="echoid-s10698" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10699" xml:space="preserve">QS* QR* QI. </s>
  <s xml:id="echoid-s10700" xml:space="preserve">\\ QL* QK* QI. </s>
  <s xml:id="echoid-s10701" xml:space="preserve">\\ PG* PF* PE. </s>
  <s xml:id="echoid-s10702" xml:space="preserve">\\ PE* PE* PE.</s>
  <s xml:id="echoid-s10703" xml:space="preserve">} Sunt {.</s>
  <s xml:id="echoid-s10704" xml:space="preserve">./.</s>
  <s xml:id="echoid-s10705" xml:space="preserve">.}. </s>
  <s xml:id="echoid-s10706" xml:space="preserve">unde QR. </s>
  <s xml:id="echoid-s10707" xml:space="preserve">QK:</s>
  <s xml:id="echoid-s10708" xml:space="preserve">: <lb/>PF. </s>
  <s xml:id="echoid-s10709" xml:space="preserve">PE.</s>
  <s xml:id="echoid-s10710" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10711" xml:space="preserve">Not. </s>
  <s xml:id="echoid-s10712" xml:space="preserve">Pro lineis rectis AB, HE, CD ſubſtitui poſſent quælibet, <lb/>etiam curvæ, parallelæ.</s>
  <s xml:id="echoid-s10713" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10714" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s10715" xml:space="preserve">Sint rurſus, in A concurrentes duæ rectæ AB, AD, rectaq; <lb/></s>
  <s xml:id="echoid-s10716" xml:space="preserve">
<anchor type="note" xlink:label="note-0236-02a" xlink:href="note-0236-02"/>
BD poſitione data; </s>
  <s xml:id="echoid-s10717" xml:space="preserve">item duæ curvæ EBE, FBF ſic relatæ, ut ductâ <lb/>utcunque PG ad DB parallelâ, ſit ſemper PF eodem ordine media <lb/>proportionalis inter PG, PE; </s>
  <s xml:id="echoid-s10718" xml:space="preserve">tum connexâ AE, ſit alia cu<unsure/>rva <lb/>KEK talis, ut ductâ quâpiam rectâ QLI ad DB parallelâ ſit ſemper <lb/>QK eodem ordine media inter QL, QI, quo fuit PF inter P G, PE; <lb/></s>
  <s xml:id="echoid-s10719" xml:space="preserve">erit rurſus linea FEF ipſi KBK analoga; </s>
  <s xml:id="echoid-s10720" xml:space="preserve">ſeu perpetim QR. </s>
  <s xml:id="echoid-s10721" xml:space="preserve">QK :</s>
  <s xml:id="echoid-s10722" xml:space="preserve">: <lb/>PF. </s>
  <s xml:id="echoid-s10723" xml:space="preserve">PE.</s>
  <s xml:id="echoid-s10724" xml:space="preserve"/>
</p>
<div xml:id="echoid-div289" type="float" level="2" n="7">
<note position="left" xlink:label="note-0236-02" xlink:href="note-0236-02a" xml:space="preserve">Fig. 66.</note>
</div>
<p>
  <s xml:id="echoid-s10725" xml:space="preserve">Nam QS* QR* QI. </s>
  <s xml:id="echoid-s10726" xml:space="preserve">\\ QL* QK* QI. </s>
  <s xml:id="echoid-s10727" xml:space="preserve">\\ PG* PF* PE. </s>
  <s xml:id="echoid-s10728" xml:space="preserve">\\ PE* PE* PE.</s>
  <s xml:id="echoid-s10729" xml:space="preserve">} ſunt {.</s>
  <s xml:id="echoid-s10730" xml:space="preserve">./.</s>
  <s xml:id="echoid-s10731" xml:space="preserve">.}.</s>
  <s xml:id="echoid-s10732" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10733" xml:space="preserve">item QS. </s>
  <s xml:id="echoid-s10734" xml:space="preserve">QL :</s>
  <s xml:id="echoid-s10735" xml:space="preserve">: \\ PG. </s>
  <s xml:id="echoid-s10736" xml:space="preserve">PE. </s>
  <s xml:id="echoid-s10737" xml:space="preserve">\\ Et QI. </s>
  <s xml:id="echoid-s10738" xml:space="preserve">QI :</s>
  <s xml:id="echoid-s10739" xml:space="preserve">: \\ PE. </s>
  <s xml:id="echoid-s10740" xml:space="preserve">PE.</s>
  <s xml:id="echoid-s10741" xml:space="preserve">}ergò QR. <lb/></s>
  <s xml:id="echoid-s10742" xml:space="preserve">QK :</s>
  <s xml:id="echoid-s10743" xml:space="preserve">: <lb/>PF. </s>
  <s xml:id="echoid-s10744" xml:space="preserve">PE.</s>
  <s xml:id="echoid-s10745" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10746" xml:space="preserve">_Not_. </s>
  <s xml:id="echoid-s10747" xml:space="preserve">Pro rectis AB, AH, AD ſubſtitui poſſent tres quævis lineæ <lb/>analogæ.</s>
  <s xml:id="echoid-s10748" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10749" xml:space="preserve">XI Item, ſit circulus AG B, cujus centrum D; </s>
  <s xml:id="echoid-s10750" xml:space="preserve">aliæque duæ curvæ <lb/>EBE, FBF tales, utper D ductâ quâcunque rectâ DG, ſit perpe-<lb/>
<anchor type="note" xlink:label="note-0236-03a" xlink:href="note-0236-03"/>
tuò DF eodem ordine media propor<unsure/>ionalis inter DG, DE; </s>
  <s xml:id="echoid-s10751" xml:space="preserve">tum <lb/>centro D per E deſ@ribatur circulus H E; </s>
  <s xml:id="echoid-s10752" xml:space="preserve">ſitque præterea curva KEK <lb/>talis, ut ductâ per D quâpiam (ad circulum HE) rectâ DL, ſit ſemper
<pb o="59" file="0237" n="252" rhead=""/>
DK eodem ordine media inter DL, DI, quo fuerat DF inter DG, <lb/>DE; </s>
  <s xml:id="echoid-s10753" xml:space="preserve">erunt curvæ FBF, KBK analogæ, ſeu perpetuò DR. </s>
  <s xml:id="echoid-s10754" xml:space="preserve">DK :</s>
  <s xml:id="echoid-s10755" xml:space="preserve">: <lb/>DF. </s>
  <s xml:id="echoid-s10756" xml:space="preserve">DE.</s>
  <s xml:id="echoid-s10757" xml:space="preserve"/>
</p>
<div xml:id="echoid-div290" type="float" level="2" n="8">
<note position="left" xlink:label="note-0236-03" xlink:href="note-0236-03a" xml:space="preserve">Fig. 67.</note>
</div>
<p>
  <s xml:id="echoid-s10758" xml:space="preserve">Nam rurſus DS.</s>
  <s xml:id="echoid-s10759" xml:space="preserve">* DR.</s>
  <s xml:id="echoid-s10760" xml:space="preserve">* DI. </s>
  <s xml:id="echoid-s10761" xml:space="preserve">\\ DL.</s>
  <s xml:id="echoid-s10762" xml:space="preserve">* DK.</s>
  <s xml:id="echoid-s10763" xml:space="preserve">* DI. </s>
  <s xml:id="echoid-s10764" xml:space="preserve">\\ DG.</s>
  <s xml:id="echoid-s10765" xml:space="preserve">*DF.</s>
  <s xml:id="echoid-s10766" xml:space="preserve">* DE \\ DE.</s>
  <s xml:id="echoid-s10767" xml:space="preserve">* DE.</s>
  <s xml:id="echoid-s10768" xml:space="preserve">* DE.</s>
  <s xml:id="echoid-s10769" xml:space="preserve">} ſunt {.</s>
  <s xml:id="echoid-s10770" xml:space="preserve">./.</s>
  <s xml:id="echoid-s10771" xml:space="preserve">.}.</s>
  <s xml:id="echoid-s10772" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10773" xml:space="preserve">unde DR. </s>
  <s xml:id="echoid-s10774" xml:space="preserve">DK :</s>
  <s xml:id="echoid-s10775" xml:space="preserve">: <lb/>DF. </s>
  <s xml:id="echoid-s10776" xml:space="preserve">DE.</s>
  <s xml:id="echoid-s10777" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10778" xml:space="preserve">Rurſus, pro circulis aliæ lineæ parallelæ, vel analogæ ſubſtitui <lb/>poſſent.</s>
  <s xml:id="echoid-s10779" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10780" xml:space="preserve">X. </s>
  <s xml:id="echoid-s10781" xml:space="preserve">Sint denuò duæ lineæ quævis A GBG, EBE; </s>
  <s xml:id="echoid-s10782" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10783" xml:space="preserve">altera <lb/>FBF ſic ad iſtas relata, ut ductâ utcunque à deſignaro puncto D recta <lb/>DG, ſit perpetuò DF eodem ordine media proportionalis inter DG, <lb/>DE; </s>
  <s xml:id="echoid-s10784" xml:space="preserve">tum adſumatur linea HEL lineæ AGB analoga (ſeu talis, ut <lb/>per D utcunque ductâ DLS, ſint perpetuò DS, DL in eadem ratio-<lb/>ne) ſit denuò linea KEK talis, ut ductâ utcunque DL, ſit perpetuò <lb/>DK eodem ordine media inter DL, DI, quo priùs DF inter DG, <lb/>DE; </s>
  <s xml:id="echoid-s10785" xml:space="preserve">erit itidem linea FBF lineæ K EK analoga.</s>
  <s xml:id="echoid-s10786" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10787" xml:space="preserve">Rurſus enim DS.</s>
  <s xml:id="echoid-s10788" xml:space="preserve">* DR* DI. </s>
  <s xml:id="echoid-s10789" xml:space="preserve">\\ DL.</s>
  <s xml:id="echoid-s10790" xml:space="preserve">* DK* DI. </s>
  <s xml:id="echoid-s10791" xml:space="preserve">\\ DG.</s>
  <s xml:id="echoid-s10792" xml:space="preserve">* DF* DE \\ DE.</s>
  <s xml:id="echoid-s10793" xml:space="preserve">* DE* DE} ſunt {.</s>
  <s xml:id="echoid-s10794" xml:space="preserve">./.</s>
  <s xml:id="echoid-s10795" xml:space="preserve">.};</s>
  <s xml:id="echoid-s10796" xml:space="preserve">}Et tam primi quàm ulti-<lb/>mi quatuor termini <lb/>ſunt proportionales. <lb/></s>
  <s xml:id="echoid-s10797" xml:space="preserve">Unde liquet Propoſi-<lb/>tum.</s>
  <s xml:id="echoid-s10798" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10799" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s10800" xml:space="preserve">Sit Arithmeticè proportionalium Series A. </s>
  <s xml:id="echoid-s10801" xml:space="preserve">B. </s>
  <s xml:id="echoid-s10802" xml:space="preserve">C. </s>
  <s xml:id="echoid-s10803" xml:space="preserve">D. </s>
  <s xml:id="echoid-s10804" xml:space="preserve">E. </s>
  <s xml:id="echoid-s10805" xml:space="preserve">F; </s>
  <s xml:id="echoid-s10806" xml:space="preserve">in <lb/>qua ſumptis quibuſcunque duobus terminis D, F; </s>
  <s xml:id="echoid-s10807" xml:space="preserve">ſit terminorum à <lb/>primo A (excluſivè) ad ipſum D numerus, N; </s>
  <s xml:id="echoid-s10808" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10809" xml:space="preserve">terminorum ab A <lb/>(itidem excluſivè) ad F, ſit numerus M; </s>
  <s xml:id="echoid-s10810" xml:space="preserve">erit A -: </s>
  <s xml:id="echoid-s10811" xml:space="preserve">D. </s>
  <s xml:id="echoid-s10812" xml:space="preserve">A -: </s>
  <s xml:id="echoid-s10813" xml:space="preserve">F :</s>
  <s xml:id="echoid-s10814" xml:space="preserve">: <lb/>N. </s>
  <s xml:id="echoid-s10815" xml:space="preserve">M.</s>
  <s xml:id="echoid-s10816" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10817" xml:space="preserve">Nam eſto differentia communis, X. </s>
  <s xml:id="echoid-s10818" xml:space="preserve">eſt ergò D = A ± NX. </s>
  <s xml:id="echoid-s10819" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10820" xml:space="preserve"><lb/>F = A ± MX. </s>
  <s xml:id="echoid-s10821" xml:space="preserve">quare A -: </s>
  <s xml:id="echoid-s10822" xml:space="preserve">D = NX. </s>
  <s xml:id="echoid-s10823" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10824" xml:space="preserve">A -: </s>
  <s xml:id="echoid-s10825" xml:space="preserve">F = MX. <lb/></s>
  <s xml:id="echoid-s10826" xml:space="preserve">unde A -: </s>
  <s xml:id="echoid-s10827" xml:space="preserve">D. </s>
  <s xml:id="echoid-s10828" xml:space="preserve">A -: </s>
  <s xml:id="echoid-s10829" xml:space="preserve">F :</s>
  <s xml:id="echoid-s10830" xml:space="preserve">: ( NX. </s>
  <s xml:id="echoid-s10831" xml:space="preserve">MX:</s>
  <s xml:id="echoid-s10832" xml:space="preserve">:) N. </s>
  <s xml:id="echoid-s10833" xml:space="preserve">M.</s>
  <s xml:id="echoid-s10834" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10835" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s10836" xml:space="preserve">Hinc, ſi duæ fuerint ejuſmodi ſeries; </s>
  <s xml:id="echoid-s10837" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10838" xml:space="preserve">in utraque ſumantur
<pb o="60" file="0238" n="253" rhead=""/>
bini, eodem ordine ſibi reſpondentes, termini (puta D, Fin prima, &amp; </s>
  <s xml:id="echoid-s10839" xml:space="preserve"><lb/>P, R in ſecunda) erit A -: </s>
  <s xml:id="echoid-s10840" xml:space="preserve">D. </s>
  <s xml:id="echoid-s10841" xml:space="preserve">A -: </s>
  <s xml:id="echoid-s10842" xml:space="preserve">F :</s>
  <s xml:id="echoid-s10843" xml:space="preserve">: M -: </s>
  <s xml:id="echoid-s10844" xml:space="preserve">P. </s>
  <s xml:id="echoid-s10845" xml:space="preserve">M -: </s>
  <s xml:id="echoid-s10846" xml:space="preserve">R.</s>
  <s xml:id="echoid-s10847" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve"> <lb/>A. # B. # C. # D. # E. # F. <lb/>M. # N. # O. # P. # Q. # R. <lb/></note>
<p>
  <s xml:id="echoid-s10848" xml:space="preserve">Nam harum rationum utraque par eſt illi, quam habent ad ſe nume-<lb/>ri N, M, quales in præcedente deſignati ſunt.</s>
  <s xml:id="echoid-s10849" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10850" xml:space="preserve">Hi verò Numeri N, M vulgò terminorum, quibus aptantur, expo-<lb/>nentes, aut Indices vocantur, in ſerie quavis proportionalium; </s>
  <s xml:id="echoid-s10851" xml:space="preserve">quales <lb/>nos ſemper in ſequentibus intelligimus, ubi literas has adhibemus.</s>
  <s xml:id="echoid-s10852" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10853" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s10854" xml:space="preserve">Sint quælibet quanta A, B, C, D, E, F continuè propor-<lb/>tionalia Arithmeticè; </s>
  <s xml:id="echoid-s10855" xml:space="preserve">nec non alia totidem, ab eodem termino A in-<lb/>cipientia, Geometricè proportionalia; </s>
  <s xml:id="echoid-s10856" xml:space="preserve">ſit au-<lb/>tem illorum ſecundum B non majus horum ſe-<lb/>cundo M; </s>
  <s xml:id="echoid-s10857" xml:space="preserve">erit quodlibet in ſerie Geometrica <lb/>majus eo, quod ipſi coordinatur in ſerie Arith-<lb/>metica.</s>
  <s xml:id="echoid-s10858" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10859" xml:space="preserve">A. </s>
  <s xml:id="echoid-s10860" xml:space="preserve">B. </s>
  <s xml:id="echoid-s10861" xml:space="preserve">C. </s>
  <s xml:id="echoid-s10862" xml:space="preserve">D. </s>
  <s xml:id="echoid-s10863" xml:space="preserve">E. </s>
  <s xml:id="echoid-s10864" xml:space="preserve">F.</s>
  <s xml:id="echoid-s10865" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10866" xml:space="preserve">A. </s>
  <s xml:id="echoid-s10867" xml:space="preserve">M. </s>
  <s xml:id="echoid-s10868" xml:space="preserve">N. </s>
  <s xml:id="echoid-s10869" xml:space="preserve">O. </s>
  <s xml:id="echoid-s10870" xml:space="preserve">P. </s>
  <s xml:id="echoid-s10871" xml:space="preserve">Q.</s>
  <s xml:id="echoid-s10872" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10873" xml:space="preserve">Eſt enim A + N &amp;</s>
  <s xml:id="echoid-s10874" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10875" xml:space="preserve">2 M (vel &amp;</s>
  <s xml:id="echoid-s10876" xml:space="preserve">gt;) </s>
  <s xml:id="echoid-s10877" xml:space="preserve">2 B = A + C. </s>
  <s xml:id="echoid-s10878" xml:space="preserve">ergò N &amp;</s>
  <s xml:id="echoid-s10879" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10880" xml:space="preserve">C. <lb/></s>
  <s xml:id="echoid-s10881" xml:space="preserve">unde M + N &amp;</s>
  <s xml:id="echoid-s10882" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10883" xml:space="preserve">B + C = A + D. </s>
  <s xml:id="echoid-s10884" xml:space="preserve">Eſt autem A + O &amp;</s>
  <s xml:id="echoid-s10885" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10886" xml:space="preserve">M + <lb/>N. </s>
  <s xml:id="echoid-s10887" xml:space="preserve">ergò A + O &amp;</s>
  <s xml:id="echoid-s10888" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10889" xml:space="preserve">A + D. </s>
  <s xml:id="echoid-s10890" xml:space="preserve">Et ideò O &amp;</s>
  <s xml:id="echoid-s10891" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10892" xml:space="preserve">D. </s>
  <s xml:id="echoid-s10893" xml:space="preserve">ergò M + O &amp;</s>
  <s xml:id="echoid-s10894" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10895" xml:space="preserve"><lb/>B + D = A + E. </s>
  <s xml:id="echoid-s10896" xml:space="preserve">Eſt autem A + P &amp;</s>
  <s xml:id="echoid-s10897" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10898" xml:space="preserve">M + O. </s>
  <s xml:id="echoid-s10899" xml:space="preserve">ergò A + P <lb/>&amp;</s>
  <s xml:id="echoid-s10900" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10901" xml:space="preserve">A + E; </s>
  <s xml:id="echoid-s10902" xml:space="preserve">adeóque P &amp;</s>
  <s xml:id="echoid-s10903" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10904" xml:space="preserve">E. </s>
  <s xml:id="echoid-s10905" xml:space="preserve">ſimilique porrò diſcurſu quoad <lb/>velis.</s>
  <s xml:id="echoid-s10906" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10907" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s10908" xml:space="preserve">Hinc, ſi rurſus fuerint A, B, C, D, E, F {.</s>
  <s xml:id="echoid-s10909" xml:space="preserve">./.</s>
  <s xml:id="echoid-s10910" xml:space="preserve">.} Arithmeticè; <lb/></s>
  <s xml:id="echoid-s10911" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10912" xml:space="preserve">A, M, N, O, P, Q ſint {.</s>
  <s xml:id="echoid-s10913" xml:space="preserve">./.</s>
  <s xml:id="echoid-s10914" xml:space="preserve">.} Geometricè; </s>
  <s xml:id="echoid-s10915" xml:space="preserve">sítque ultimum F non <lb/>minus ultimo Q; </s>
  <s xml:id="echoid-s10916" xml:space="preserve">erit B majus quàm M.</s>
  <s xml:id="echoid-s10917" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10918" xml:space="preserve">Nam ſi dicatur B non majus quàm M; </s>
  <s xml:id="echoid-s10919" xml:space="preserve">erit indè F minus, quàm <lb/>Q contra hypotheſin.</s>
  <s xml:id="echoid-s10920" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10921" xml:space="preserve">Item, iiſdem poſitis; </s>
  <s xml:id="echoid-s10922" xml:space="preserve">erit penultimum E majus penultimo P.</s>
  <s xml:id="echoid-s10923" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10924" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s10925" xml:space="preserve">Nam ſi F = Q; </s>
  <s xml:id="echoid-s10926" xml:space="preserve">conſtat ex præcedente fore E &amp;</s>
  <s xml:id="echoid-s10927" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s10928" xml:space="preserve">P (ſcilicet u-<lb/>tramque ſeriem invertendo) ſin F &amp;</s>
  <s xml:id="echoid-s10929" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10930" xml:space="preserve">Q: </s>
  <s xml:id="echoid-s10931" xml:space="preserve">potiori jure liquet fore <lb/>E &amp;</s>
  <s xml:id="echoid-s10932" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10933" xml:space="preserve">P.</s>
  <s xml:id="echoid-s10934" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10935" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s10936" xml:space="preserve">Quinimò demùm, iiſdem poſitis, quodlibet in ſerie Arith-<lb/>metica majus eſt coordinato quolibet in ſerie Geometrica; </s>
  <s xml:id="echoid-s10937" xml:space="preserve">puta, C <lb/>majus eſt quàm N.</s>
  <s xml:id="echoid-s10938" xml:space="preserve"/>
</p>
<pb o="61" file="0239" n="254" rhead=""/>
<p>
  <s xml:id="echoid-s10939" xml:space="preserve">Eſt enim E &amp;</s>
  <s xml:id="echoid-s10940" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10941" xml:space="preserve">P, ac indè D &amp;</s>
  <s xml:id="echoid-s10942" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10943" xml:space="preserve">O; </s>
  <s xml:id="echoid-s10944" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10945" xml:space="preserve">hinc C &amp;</s>
  <s xml:id="echoid-s10946" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10947" xml:space="preserve">N.</s>
  <s xml:id="echoid-s10948" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10949" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s10950" xml:space="preserve">Conſectatur hinc; </s>
  <s xml:id="echoid-s10951" xml:space="preserve">ſi fuerint quatuor lineæ HBH, GBG, FBF, <lb/>EBE ſeſe interſecantes in B, ac ita verſus ſe relatæ, ut ductâ utcunque <lb/>rectâ DH ad poſitione datam DB parallelâ (in linea nempe DD D <lb/>
<anchor type="note" xlink:label="note-0239-01a" xlink:href="note-0239-01"/>
terminatâ) vel à deſignato puncto D projectâ DH; </s>
  <s xml:id="echoid-s10952" xml:space="preserve">ſit perpetuò DG <lb/>inter DH, DE eodem ordine media proportionalis Arithmeticè, quo <lb/>DF inter eaſdem media Geometricè; </s>
  <s xml:id="echoid-s10953" xml:space="preserve">lineæ GB G, FBF ſeſe mu-<lb/>tuò contingunt.</s>
  <s xml:id="echoid-s10954" xml:space="preserve"/>
</p>
<div xml:id="echoid-div291" type="float" level="2" n="9">
<note position="right" xlink:label="note-0239-01" xlink:href="note-0239-01a" xml:space="preserve">Fig. 68. <lb/>69.</note>
</div>
<p>
  <s xml:id="echoid-s10955" xml:space="preserve">Enimverò linea GBH extra lineam FBF totam cadere manifeſtum <lb/>è præcedente.</s>
  <s xml:id="echoid-s10956" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s10957" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s10958" xml:space="preserve">Ex iſthinc etiam (quod ſtrictim tranſcurrens moneo) di-<lb/>
<anchor type="note" xlink:label="note-0239-02a" xlink:href="note-0239-02"/>
verſis innumeris _Hyperbolarum_, aut _Hyperboliformium_ generibus con-<lb/>venientes rectæ ασνμπωτοι definiuntur. </s>
  <s xml:id="echoid-s10959" xml:space="preserve">Sint nempe rectæ VD, BD <lb/>poſitione datæ; </s>
  <s xml:id="echoid-s10960" xml:space="preserve">ſint item aliæ duæ rectæ AB, VI; </s>
  <s xml:id="echoid-s10961" xml:space="preserve">ductâ verò li-<lb/>berè rectâ PG ad DB parallela, ſit P φ conſtantèr inter PG, PE eo-<lb/>dem ordine media proportionalis Arithmeticè, quo PF inter eaſdem <lb/>media Geometricè; </s>
  <s xml:id="echoid-s10962" xml:space="preserve">quia jam <anchor type="note" xlink:href="" symbol="(a)"/> rectæ E G, E φ ſemper eandem ob-
<anchor type="note" xlink:label="note-0239-03a" xlink:href="note-0239-03"/>
tinent rationem, eſt linea φ φ φ recta; </s>
  <s xml:id="echoid-s10963" xml:space="preserve">verùm linea VFF eſt _hyperbo-_ <lb/>_la,_ vel _hyperboliformis_ aliqua (communis quidem vel _Apolloniana_ <lb/>_hyperbola_, ſi PF ſit inter ipſas PG, PE ſimpliciter media, ſed alia <lb/>diverſi generis quædam _hyperboliformis_, ſi PE ſit alterius cujuſpiam <lb/>ordinis media) atqui patet è penultima præmiſſa lineam φ φ φ eodem <lb/>ordine reſpondenti lineæ VFF _aſymptoton_ eſſe. </s>
  <s xml:id="echoid-s10964" xml:space="preserve">quod an πρ@@γ@ ſit <lb/>neſcio, nobis certè πáρε@γον fuit, hic adnotâſſe.</s>
  <s xml:id="echoid-s10965" xml:space="preserve"/>
</p>
<div xml:id="echoid-div292" type="float" level="2" n="10">
<note position="right" xlink:label="note-0239-02" xlink:href="note-0239-02a" xml:space="preserve">Fig. 70.</note>
<note symbol="(a)" position="right" xlink:label="note-0239-03" xlink:href="note-0239-03a" xml:space="preserve">12 buj@@</note>
</div>
<p>
  <s xml:id="echoid-s10966" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s10967" xml:space="preserve">A puncto aſſignato B ad datam poſitione rectam AC ductæ <lb/>ſint restæ tres BA, BC, BQ; </s>
  <s xml:id="echoid-s10968" xml:space="preserve">tum in QC producta ſumatur ſumptum <lb/>
<anchor type="note" xlink:label="note-0239-04a" xlink:href="note-0239-04"/>
quodpiam D; </s>
  <s xml:id="echoid-s10969" xml:space="preserve">per B recta (puta B R) duci poteſt (ad alterutras ipſi-<lb/>us BQ partes) tali@, ut à D projectâ quâcunque rectâ, ceu DN; </s>
  <s xml:id="echoid-s10970" xml:space="preserve">ſit <lb/>hujus à rectis B Q, BR intercepta pars (F E) minor ejuſdem à rectis <lb/>BA, BC interceptâ parte (N M).</s>
  <s xml:id="echoid-s10971" xml:space="preserve"/>
</p>
<div xml:id="echoid-div293" type="float" level="2" n="11">
<note position="right" xlink:label="note-0239-04" xlink:href="note-0239-04a" xml:space="preserve">Fig. 71.</note>
</div>
<p>
  <s xml:id="echoid-s10972" xml:space="preserve">Nam, primò, ſi BR (ultra angulum ABC jaceat reſpectu puncti <lb/>D; </s>
  <s xml:id="echoid-s10973" xml:space="preserve">ſiat QR = CA; </s>
  <s xml:id="echoid-s10974" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10975" xml:space="preserve">connectatur BR; </s>
  <s xml:id="echoid-s10976" xml:space="preserve">tum utcunque ducatur <lb/>DE, rectas ſecans, ut vides; </s>
  <s xml:id="echoid-s10977" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10978" xml:space="preserve">maniſeſtum eſt, * è ſupra mon-<lb/>
<anchor type="note" xlink:label="note-0239-05a" xlink:href="note-0239-05"/>
ſtratis fore, FE &amp;</s>
  <s xml:id="echoid-s10979" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s10980" xml:space="preserve">NM.</s>
  <s xml:id="echoid-s10981" xml:space="preserve"/>
</p>
<div xml:id="echoid-div294" type="float" level="2" n="12">
<note position="right" xlink:label="note-0239-05" xlink:href="note-0239-05a" xml:space="preserve">* Per 7. Lect. <lb/>VI.</note>
</div>
<p>
  <s xml:id="echoid-s10982" xml:space="preserve">Sin B Q citra angulum ABC cadat verſus D; </s>
  <s xml:id="echoid-s10983" xml:space="preserve">(a) ducatur recta <lb/>
<anchor type="note" xlink:label="note-0239-06a" xlink:href="note-0239-06"/>
BH talis, ut à BQ, B H interceptæ minores ſint interceptis à BQ, <lb/>BA; </s>
  <s xml:id="echoid-s10984" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10985" xml:space="preserve">ſumatur HR = QC; </s>
  <s xml:id="echoid-s10986" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10987" xml:space="preserve">connectatur BR; </s>
  <s xml:id="echoid-s10988" xml:space="preserve">tum rurſus <lb/>
<anchor type="note" xlink:label="note-0239-07a" xlink:href="note-0239-07"/>
utcunque ductâ DN, quæ rectas interſecet, ut exhibet Schema; </s>
  <s xml:id="echoid-s10989" xml:space="preserve">quo-
<pb o="62" file="0240" n="255" rhead=""/>
niam jam eſt KF <anchor type="note" xlink:href="" symbol="(b)"/> &amp;</s>
  <s xml:id="echoid-s10990" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s10991" xml:space="preserve">NF; </s>
  <s xml:id="echoid-s10992" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s10993" xml:space="preserve">KE * &amp;</s>
  <s xml:id="echoid-s10994" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s10995" xml:space="preserve">MF; </s>
  <s xml:id="echoid-s10996" xml:space="preserve">perſpicuum eſt
<anchor type="note" xlink:label="note-0240-01a" xlink:href="note-0240-01"/>
reſtare FE &amp;</s>
  <s xml:id="echoid-s10997" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s10998" xml:space="preserve">NM.</s>
  <s xml:id="echoid-s10999" xml:space="preserve"/>
</p>
<div xml:id="echoid-div295" type="float" level="2" n="13">
<note position="right" xlink:label="note-0239-06" xlink:href="note-0239-06a" xml:space="preserve">* Per <lb/>Vi. 8 Lect.</note>
<note position="right" xlink:label="note-0239-07" xlink:href="note-0239-07a" xml:space="preserve">Fig. 72.</note>
<note position="left" xlink:label="note-0240-01" xlink:href="note-0240-01a" xml:space="preserve"> (b) _Conſtr._</note>
</div>
<p>
  <s xml:id="echoid-s11000" xml:space="preserve">Ità quidem ab una rectæ BQ parte recta BR duci poteſt, quæ mi-<lb/>nores ipſis MN intercipiat; </s>
  <s xml:id="echoid-s11001" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(a)"/> poteſt autem ab altera parte recta
quoque duci, quæ minores intercipiat ipſis F E; </s>
  <s xml:id="echoid-s11002" xml:space="preserve">unde totum liquet <lb/>Propoſitum.</s>
  <s xml:id="echoid-s11003" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11004" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s11005" xml:space="preserve">In recta DZ ſint tria puncta D, E, F; </s>
  <s xml:id="echoid-s11006" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11007" xml:space="preserve">in F ſit vertex an-<lb/>
<anchor type="note" xlink:label="note-0240-02a" xlink:href="note-0240-02"/>
guli rectilinei BFC, cujus latera ſecet recta DBC; </s>
  <s xml:id="echoid-s11008" xml:space="preserve">per E vero <lb/>ducta ſit recta EG; </s>
  <s xml:id="echoid-s11009" xml:space="preserve">poteſt ab E recta duci (ceu EH) talis, ut à <lb/>puncto D projectâ utcunque rectâ DK ſit in hac à rectis EG, EH in-<lb/>tercepta minor à rectis FC, FB interceptâ.</s>
  <s xml:id="echoid-s11010" xml:space="preserve"/>
</p>
<div xml:id="echoid-div296" type="float" level="2" n="14">
<note position="left" xlink:label="note-0240-02" xlink:href="note-0240-02a" xml:space="preserve">Fig. 73.</note>
</div>
<p>
  <s xml:id="echoid-s11011" xml:space="preserve">Ducantur ES ad FC, &amp; </s>
  <s xml:id="echoid-s11012" xml:space="preserve">ER ad FB parallelæ; </s>
  <s xml:id="echoid-s11013" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11014" xml:space="preserve">in primo caſu, <lb/>ubi punctum E puncto D vicinius eſt, (ob ſimilitudinem triangulorum <lb/>ENM, FKI) manifeſtum eſt fore MN &amp;</s>
  <s xml:id="echoid-s11015" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11016" xml:space="preserve">IK; </s>
  <s xml:id="echoid-s11017" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(a)"/> poteſt au-
<anchor type="note" xlink:label="note-0240-03a" xlink:href="note-0240-03"/>
tem ab E duci recta (puta EH) talis, ut interceptæ PO minores ſint <lb/>interceptis MN; </s>
  <s xml:id="echoid-s11018" xml:space="preserve">ergò liquet.</s>
  <s xml:id="echoid-s11019" xml:space="preserve"/>
</p>
<div xml:id="echoid-div297" type="float" level="2" n="15">
<note position="left" xlink:label="note-0240-03" xlink:href="note-0240-03a" xml:space="preserve">(a) _19. bujus._</note>
</div>
<p>
  <s xml:id="echoid-s11020" xml:space="preserve">In altero caſu, ubi punctum F ipſi D propius, ſumatur SL æqualis <lb/>ipſi CB; </s>
  <s xml:id="echoid-s11021" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11022" xml:space="preserve">connectatur EL; </s>
  <s xml:id="echoid-s11023" xml:space="preserve">Eſtque jam IK. </s>
  <s xml:id="echoid-s11024" xml:space="preserve">MN :</s>
  <s xml:id="echoid-s11025" xml:space="preserve">: FK. </s>
  <s xml:id="echoid-s11026" xml:space="preserve">EN :</s>
  <s xml:id="echoid-s11027" xml:space="preserve">: <lb/>
<anchor type="note" xlink:label="note-0240-04a" xlink:href="note-0240-04"/>
DF. </s>
  <s xml:id="echoid-s11028" xml:space="preserve">DE :</s>
  <s xml:id="echoid-s11029" xml:space="preserve">: FC. </s>
  <s xml:id="echoid-s11030" xml:space="preserve">ES :</s>
  <s xml:id="echoid-s11031" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s11032" xml:space="preserve">RS <anchor type="note" xlink:href="" symbol="(c)"/>:</s>
  <s xml:id="echoid-s11033" xml:space="preserve">: LS. </s>
  <s xml:id="echoid-s11034" xml:space="preserve">RS <anchor type="note" xlink:href="" symbol="(d)"/> &amp;</s>
  <s xml:id="echoid-s11035" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11036" xml:space="preserve">QN.</s>
  <s xml:id="echoid-s11037" xml:space="preserve">
<anchor type="note" xlink:label="note-0240-05a" xlink:href="note-0240-05"/>
MN. </s>
  <s xml:id="echoid-s11038" xml:space="preserve">quapropter eſt IK &amp;</s>
  <s xml:id="echoid-s11039" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11040" xml:space="preserve">QN. </s>
  <s xml:id="echoid-s11041" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(a)"/> poteſt autem ab E recta duci,
<anchor type="note" xlink:label="note-0240-06a" xlink:href="note-0240-06"/>
ceu E H, ſic ut ab EG, EH interceptæ OP minores ſint interceptis <lb/>QN. </s>
  <s xml:id="echoid-s11042" xml:space="preserve">quamobrem abundè conſtat Propoſitum.</s>
  <s xml:id="echoid-s11043" xml:space="preserve"/>
</p>
<div xml:id="echoid-div298" type="float" level="2" n="16">
<note position="left" xlink:label="note-0240-04" xlink:href="note-0240-04a" xml:space="preserve">Fig. 74.</note>
<note position="left" xlink:label="note-0240-05" xlink:href="note-0240-05a" xml:space="preserve">(_c_) _Conſtr_.</note>
<note position="left" xlink:label="note-0240-06" xlink:href="note-0240-06a" xml:space="preserve">(_d_)6. Lect. VI.</note>
</div>
<p>
  <s xml:id="echoid-s11044" xml:space="preserve">XXI Curvam BA tangat recta BO in B; </s>
  <s xml:id="echoid-s11045" xml:space="preserve">ſitque recta BO æ-<lb/>qualis curvæ B A; </s>
  <s xml:id="echoid-s11046" xml:space="preserve">ſumpto tunc in curva puncto quopiam K conne-<lb/>
<anchor type="note" xlink:label="note-0240-07a" xlink:href="note-0240-07"/>
ctatur recta KO; </s>
  <s xml:id="echoid-s11047" xml:space="preserve">erit KO major arcu KA.</s>
  <s xml:id="echoid-s11048" xml:space="preserve"/>
</p>
<div xml:id="echoid-div299" type="float" level="2" n="17">
<note position="left" xlink:label="note-0240-07" xlink:href="note-0240-07a" xml:space="preserve">Fig. 75.</note>
</div>
<p>
  <s xml:id="echoid-s11049" xml:space="preserve">Nam, quoniam recta minimum eſt inter bina puncta intervallum, <lb/>eſt BK + KO &amp;</s>
  <s xml:id="echoid-s11050" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11051" xml:space="preserve">BO = BK + KA. </s>
  <s xml:id="echoid-s11052" xml:space="preserve">ergò KA &amp;</s>
  <s xml:id="echoid-s11053" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11054" xml:space="preserve">KO.</s>
  <s xml:id="echoid-s11055" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11056" xml:space="preserve">XXII. </s>
  <s xml:id="echoid-s11057" xml:space="preserve">Hinc, utcunque ſumptis (ad eaſdem contactûs partes) duobus <lb/>punctis K, L, connexâque rectâ KL; </s>
  <s xml:id="echoid-s11058" xml:space="preserve">erit KL + LO &amp;</s>
  <s xml:id="echoid-s11059" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11060" xml:space="preserve">KA.</s>
  <s xml:id="echoid-s11061" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11062" xml:space="preserve">Nam, ſupra contactum verſus A, eſt KL + LO &amp;</s>
  <s xml:id="echoid-s11063" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11064" xml:space="preserve">KO &amp;</s>
  <s xml:id="echoid-s11065" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11066" xml:space="preserve">KA.</s>
  <s xml:id="echoid-s11067" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11068" xml:space="preserve">Infra verò, eſt KL + LB &amp;</s>
  <s xml:id="echoid-s11069" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11070" xml:space="preserve">KB (ex hypotheſibus _Archime-_ <lb/>_dæis_) adeóque KL + LO &amp;</s>
  <s xml:id="echoid-s11071" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11072" xml:space="preserve">KA.</s>
  <s xml:id="echoid-s11073" xml:space="preserve"/>
</p>
<pb o="63" file="0241" n="256"/>
</div>
<div xml:id="echoid-div301" type="section" level="1" n="33">
<head xml:id="echoid-head36" xml:space="preserve"><emph style="sc">Lect</emph>. VIII.</head>
<p>
  <s xml:id="echoid-s11074" xml:space="preserve">MIhi ſanè videor ( videbor &amp; </s>
  <s xml:id="echoid-s11075" xml:space="preserve">vobis, opinor ) quod irridebat <lb/>_ſapiensille Scurra, perquam exiguæ Civitati portas ingentes_ <lb/>_extrnxiſſe_ Nec enim adhuc aliud quàm ad rem aliquanto propiùs eni-<lb/>timur. </s>
  <s xml:id="echoid-s11076" xml:space="preserve">ad illam.</s>
  <s xml:id="echoid-s11077" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11078" xml:space="preserve">I. </s>
  <s xml:id="echoid-s11079" xml:space="preserve">Hæcadſumimus. </s>
  <s xml:id="echoid-s11080" xml:space="preserve">Si duæ lineæ ( OMO, TMT ) ſeſe con-<lb/>
<anchor type="note" xlink:label="note-0241-01a" xlink:href="note-0241-01"/>
tingant, angulosipſæ comprehendunt ( OMT ) rectilineo quovis an-<lb/>gulo minores. </s>
  <s xml:id="echoid-s11081" xml:space="preserve">Et vice versâ: </s>
  <s xml:id="echoid-s11082" xml:space="preserve">Si duæ lineæ ( OMO. </s>
  <s xml:id="echoid-s11083" xml:space="preserve">TMT ) an-<lb/>gulos contineant quovis rectilineo minores, illæ ſeſe contingent _(_con-<lb/>tingentibus ſaltem æquipollebunt_)_.</s>
  <s xml:id="echoid-s11084" xml:space="preserve"/>
</p>
<div xml:id="echoid-div301" type="float" level="2" n="1">
<note position="right" xlink:label="note-0241-01" xlink:href="note-0241-01a" xml:space="preserve">Fig. 76, <lb/>77.</note>
</div>
<p>
  <s xml:id="echoid-s11085" xml:space="preserve">Hujus _effati_ rationem jampridem _(_ni fallor_)_ attigimus.</s>
  <s xml:id="echoid-s11086" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11087" xml:space="preserve">II. </s>
  <s xml:id="echoid-s11088" xml:space="preserve">Hinc; </s>
  <s xml:id="echoid-s11089" xml:space="preserve">Si duas lineas OMO, TMT tertia quæpiam linea <lb/>PM P contingat, ipſæ etiam lineæ OMO, TMT ſeſe contin-<lb/>gent.</s>
  <s xml:id="echoid-s11090" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11091" xml:space="preserve">Nam quoniam lineæ OMO, PM P ſeſe contingunt, erit angulus <lb/>OM P quovis _rectilineo_ minor. </s>
  <s xml:id="echoid-s11092" xml:space="preserve">Item, ob linearum TMT, PMP <lb/>_contractum_, erit _angulus_ TM P quovis etiam _rectilineo_ minor. </s>
  <s xml:id="echoid-s11093" xml:space="preserve">Erit <lb/>igitur angulus TMO _rectilineo_ quovis minor. </s>
  <s xml:id="echoid-s11094" xml:space="preserve">Unde lineæ OMO, <lb/>TMT ſe mutuo contingent.</s>
  <s xml:id="echoid-s11095" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11096" xml:space="preserve">III. </s>
  <s xml:id="echoid-s11097" xml:space="preserve">Tangat recta FA curvam FX in F; </s>
  <s xml:id="echoid-s11098" xml:space="preserve">ſitque poſitione data recta <lb/>FE; </s>
  <s xml:id="echoid-s11099" xml:space="preserve">ſint item duæ curvæ EY, EZ tales, ut ductâ utcunque rectâ <lb/>
<anchor type="note" xlink:label="note-0241-02a" xlink:href="note-0241-02"/>
IL ad EF parallelâ ( quæ lineas expoſitas ſecet, ut vides ) ſit ſemper <lb/>intercepta KL æqualis interceptæ I G; </s>
  <s xml:id="echoid-s11100" xml:space="preserve">etiam curvæ EY, EZ ſeſe <lb/>contingent.</s>
  <s xml:id="echoid-s11101" xml:space="preserve"/>
</p>
<div xml:id="echoid-div302" type="float" level="2" n="2">
<note position="right" xlink:label="note-0241-02" xlink:href="note-0241-02a" xml:space="preserve">Fig 78.</note>
</div>
<p>
  <s xml:id="echoid-s11102" xml:space="preserve">Si non tangant, poteſt inter ipſas conſtitui angulus rectilineus, puta <lb/>BEC; </s>
  <s xml:id="echoid-s11103" xml:space="preserve">hunc utcunque ſecet ad FE parallela I L; </s>
  <s xml:id="echoid-s11104" xml:space="preserve">ſumatúrque G H <lb/>= BC, &amp; </s>
  <s xml:id="echoid-s11105" xml:space="preserve">connectatur F H; </s>
  <s xml:id="echoid-s11106" xml:space="preserve">ſunt igitur è parallelis ad FE à rectis
<pb o="64" file="0242" n="257" rhead=""/>
FG, FH interceptæ pares interceptis ab EB, EC; </s>
  <s xml:id="echoid-s11107" xml:space="preserve">hoc eſt minores <lb/>interceptis à curvis EY, EZ; </s>
  <s xml:id="echoid-s11108" xml:space="preserve">hoc eſt minores interceptis à curva <lb/>FX, &amp; </s>
  <s xml:id="echoid-s11109" xml:space="preserve">recta FA; </s>
  <s xml:id="echoid-s11110" xml:space="preserve">quapropter angulus XFA rectilineo HFG ma-<lb/>jor eſt; </s>
  <s xml:id="echoid-s11111" xml:space="preserve">unde recta FA curvam FX non tangit, contra _Hy-_ <lb/>_potheſin_.</s>
  <s xml:id="echoid-s11112" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11113" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s11114" xml:space="preserve">Itidem, Tangat recta FA curvam FX, &amp; </s>
  <s xml:id="echoid-s11115" xml:space="preserve">ſint duæ curvæ <lb/>EY, EZ tales, ut ab aſſignato puncto D utcunque ductâ rectâ IL <lb/>
<anchor type="note" xlink:label="note-0242-01a" xlink:href="note-0242-01"/>
( quæ lineas expoſitas ſecet ut vides ) ſit ſemper KL = IG; </s>
  <s xml:id="echoid-s11116" xml:space="preserve">curvæ <lb/>EY, EZ ſeſe tangent.</s>
  <s xml:id="echoid-s11117" xml:space="preserve"/>
</p>
<div xml:id="echoid-div303" type="float" level="2" n="3">
<note position="left" xlink:label="note-0242-01" xlink:href="note-0242-01a" xml:space="preserve">Fig. 79.</note>
</div>
<p>
  <s xml:id="echoid-s11118" xml:space="preserve">Nam, ſineges, his interſeratur _angulus rectiline<unsure/>us_ BEC; </s>
  <s xml:id="echoid-s11119" xml:space="preserve">quem <lb/>utcunque a D projecta ſecet recta DL; </s>
  <s xml:id="echoid-s11120" xml:space="preserve"><anchor type="note" xlink:href="" symbol="_(a)_"/> poteſt jam ab F recta du-
<anchor type="note" xlink:label="note-0242-02a" xlink:href="note-0242-02"/>
ci _(_puta FH_)_ talis, ut ſint è projectis à D a rectis FG, FH inter-<lb/>ceptæ minores interceptis abipſis EB, EC, hoc eſt multo minores in-<lb/>terceptis à recta E A, curváque FX. </s>
  <s xml:id="echoid-s11121" xml:space="preserve">Unde ſequetur angulum AFX <lb/>rectilineo GF H majorem eſſe; </s>
  <s xml:id="echoid-s11122" xml:space="preserve">ac idcircò rectam AF non conting ere <lb/>curvam FX, adverſus _Hypotheſin_.</s>
  <s xml:id="echoid-s11123" xml:space="preserve"/>
</p>
<div xml:id="echoid-div304" type="float" level="2" n="4">
<note position="left" xlink:label="note-0242-02" xlink:href="note-0242-02a" xml:space="preserve">(_a_) 20 Lect. <lb/>VII.</note>
</div>
<p>
  <s xml:id="echoid-s11124" xml:space="preserve">Hæ præcedentes duæ Concluſiones veræ ſunt, &amp; </s>
  <s xml:id="echoid-s11125" xml:space="preserve">ſimili ratione de-<lb/>monſtrantur, poſito interceptas IG, KL quamvis ad ſe perpetim ha-<lb/>bere proportionem eandem. </s>
  <s xml:id="echoid-s11126" xml:space="preserve">Parco verbis.</s>
  <s xml:id="echoid-s11127" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11128" xml:space="preserve">Propoſuimus hæc, ut ſequentium nonnulla à ſcrupulis munian-<lb/>tur.</s>
  <s xml:id="echoid-s11129" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11130" xml:space="preserve">V. </s>
  <s xml:id="echoid-s11131" xml:space="preserve">Sit recta VEI, duæque curvæ YFN, ZGO ſic ad ſe relatæ, <lb/>ut ductâ utcunque rectâ EFG ad poſitione datam AB parallelâ, ha-<lb/>beant interceptæ E G, EF ſemper eandem rationem inter ſe; </s>
  <s xml:id="echoid-s11132" xml:space="preserve">tangat <lb/>autem recta TG curvarum unam ZGO in G _(_cum recta VE con-<lb/>veniens in T _)_ ducta TF alteram YFN quoque continget.</s>
  <s xml:id="echoid-s11133" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11134" xml:space="preserve">Nam utcun que ducatur recta IL (lineas expoſitas ut vides interſe-<lb/>cans ) Eſtigitur IL. </s>
  <s xml:id="echoid-s11135" xml:space="preserve">IN <anchor type="note" xlink:href="" symbol="(_a_)"/> &amp;</s>
  <s xml:id="echoid-s11136" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11137" xml:space="preserve">IO. </s>
  <s xml:id="echoid-s11138" xml:space="preserve">IN :</s>
  <s xml:id="echoid-s11139" xml:space="preserve">: EG. </s>
  <s xml:id="echoid-s11140" xml:space="preserve">EF :</s>
  <s xml:id="echoid-s11141" xml:space="preserve">: IL. </s>
  <s xml:id="echoid-s11142" xml:space="preserve">IK.</s>
  <s xml:id="echoid-s11143" xml:space="preserve">
<anchor type="note" xlink:label="note-0242-03a" xlink:href="note-0242-03"/>
Igitur IN &amp;</s>
  <s xml:id="echoid-s11144" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11145" xml:space="preserve">IK. </s>
  <s xml:id="echoid-s11146" xml:space="preserve">ergò punctum K extra curvam YFN jacet; </s>
  <s xml:id="echoid-s11147" xml:space="preserve">totáq; <lb/></s>
  <s xml:id="echoid-s11148" xml:space="preserve">recta TF.</s>
  <s xml:id="echoid-s11149" xml:space="preserve"/>
</p>
<div xml:id="echoid-div305" type="float" level="2" n="5">
<note position="left" xlink:label="note-0242-03" xlink:href="note-0242-03a" xml:space="preserve">(_a_) _Hyp_</note>
</div>
<p>
  <s xml:id="echoid-s11150" xml:space="preserve">Aliter. </s>
  <s xml:id="echoid-s11151" xml:space="preserve">Eſt IL. </s>
  <s xml:id="echoid-s11152" xml:space="preserve">IK :</s>
  <s xml:id="echoid-s11153" xml:space="preserve">: (IO. </s>
  <s xml:id="echoid-s11154" xml:space="preserve">IN :</s>
  <s xml:id="echoid-s11155" xml:space="preserve">: IK-IO. </s>
  <s xml:id="echoid-s11156" xml:space="preserve">IK-IN :</s>
  <s xml:id="echoid-s11157" xml:space="preserve">:) <lb/>OL. </s>
  <s xml:id="echoid-s11158" xml:space="preserve">NK, ergò cùm lineæ GL, GO ſe <anchor type="note" xlink:href="" symbol="(_b_)"/> tangant, <anchor type="note" xlink:href="" symbol="(_c_)"/> etiam li-
<anchor type="note" xlink:label="note-0242-04a" xlink:href="note-0242-04"/>
neæ F N, FK ſeſe tangent.</s>
  <s xml:id="echoid-s11159" xml:space="preserve"/>
</p>
<div xml:id="echoid-div306" type="float" level="2" n="6">
<note position="left" xlink:label="note-0242-04" xlink:href="note-0242-04a" xml:space="preserve">(_b_)_Hyp_.</note>
</div>
<note position="left" xml:space="preserve">_(c)Schol. 4. bu-_ <lb/>_jus._</note>
<p>
  <s xml:id="echoid-s11160" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s11161" xml:space="preserve">Etiam ſi tres curvæ XEM, YFN, ZGO itâ referantur ad ſe, <lb/>
<anchor type="note" xlink:label="note-0242-06a" xlink:href="note-0242-06"/>
ut ductâ utcunque rectâ EFG ad poſitione datam parallelâ, ſint ſem-<lb/>per EG, EF in eadem ra ione, concurrant autem duarum XEM, <lb/>ZGO tangentes ET, GT in T; </s>
  <s xml:id="echoid-s11162" xml:space="preserve">adjuncta TF curvam YFN tan-<lb/>get.</s>
  <s xml:id="echoid-s11163" xml:space="preserve"/>
</p>
<div xml:id="echoid-div307" type="float" level="2" n="7">
<note position="left" xlink:label="note-0242-06" xlink:href="note-0242-06a" xml:space="preserve">Fig. 80.</note>
</div>
<pb o="65" file="0243" n="258" rhead=""/>
<p>
  <s xml:id="echoid-s11164" xml:space="preserve">Nam (facto ut priùs) erit IL. </s>
  <s xml:id="echoid-s11165" xml:space="preserve">IK :</s>
  <s xml:id="echoid-s11166" xml:space="preserve">: EG. </s>
  <s xml:id="echoid-s11167" xml:space="preserve">EF :</s>
  <s xml:id="echoid-s11168" xml:space="preserve">: MO. </s>
  <s xml:id="echoid-s11169" xml:space="preserve">MN.</s>
  <s xml:id="echoid-s11170" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11171" xml:space="preserve">* quapropter erit punctum K extra curvam YFN.</s>
  <s xml:id="echoid-s11172" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">* 2. Lect. VII.</note>
<p>
  <s xml:id="echoid-s11173" xml:space="preserve">Poſſit hæc, ut præceden@, aliter oſtendi; </s>
  <s xml:id="echoid-s11174" xml:space="preserve">ſed verbis pluribus.</s>
  <s xml:id="echoid-s11175" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11176" xml:space="preserve">Curvas ità ſitas concipe quales figura monſtrat. </s>
  <s xml:id="echoid-s11177" xml:space="preserve">nam {στ}ενολχίαν} ego <lb/>ac{αδολεοιαν} fugitans caſus præ cæteris obvios ac faciles arripiens pro-<lb/>pono. </s>
  <s xml:id="echoid-s11178" xml:space="preserve">Hoc ubique ſubnotatum velim.</s>
  <s xml:id="echoid-s11179" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11180" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s11181" xml:space="preserve">Sit punctum datum D, curvæque duæ XEM, YFN, ità <lb/>relatæ, ut à D projectâ quacunque rectâ DEF, habeant ad ſe rectæ <lb/>
<anchor type="note" xlink:label="note-0243-02a" xlink:href="note-0243-02"/>
DE, DF rationem ſemper eandem; </s>
  <s xml:id="echoid-s11182" xml:space="preserve">unam verò YFN tangat recta <lb/>FS; </s>
  <s xml:id="echoid-s11183" xml:space="preserve">cui parallela ſit ER; </s>
  <s xml:id="echoid-s11184" xml:space="preserve">tanget recta ER curvam XEM.</s>
  <s xml:id="echoid-s11185" xml:space="preserve"/>
</p>
<div xml:id="echoid-div308" type="float" level="2" n="8">
<note position="right" xlink:label="note-0243-02" xlink:href="note-0243-02a" xml:space="preserve">Fig. 81.</note>
</div>
<p>
  <s xml:id="echoid-s11186" xml:space="preserve">Nam à D utcunque projiciatur recta DK _(_ lineas interſecans, ut <lb/>vides). </s>
  <s xml:id="echoid-s11187" xml:space="preserve">Eſtque DK. </s>
  <s xml:id="echoid-s11188" xml:space="preserve">DI :</s>
  <s xml:id="echoid-s11189" xml:space="preserve">: DF. </s>
  <s xml:id="echoid-s11190" xml:space="preserve">DE :</s>
  <s xml:id="echoid-s11191" xml:space="preserve">: DN. </s>
  <s xml:id="echoid-s11192" xml:space="preserve">DM; </s>
  <s xml:id="echoid-s11193" xml:space="preserve">ergò quum ſit <lb/>DK &amp;</s>
  <s xml:id="echoid-s11194" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11195" xml:space="preserve">DN; </s>
  <s xml:id="echoid-s11196" xml:space="preserve">erit DI &amp;</s>
  <s xml:id="echoid-s11197" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11198" xml:space="preserve">DM; </s>
  <s xml:id="echoid-s11199" xml:space="preserve">quare tota recta RE extra curvam <lb/>XEM cadit.</s>
  <s xml:id="echoid-s11200" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11201" xml:space="preserve">Rectæ NK, MI rationem ſemper eandem obtinent; </s>
  <s xml:id="echoid-s11202" xml:space="preserve">unde res ali-<lb/>ter conſtat.</s>
  <s xml:id="echoid-s11203" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11204" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s11205" xml:space="preserve">Sint tres curvæ XEM, YFN, ZG O tales, ut ſi ab aſſig-<lb/>nato puncto D projiciatur utcunque recta DEFG, habeant interceptæ <lb/>EG, EF rationem ſemper eandem (puta quam R ad _S_) tangant au-<lb/>
<anchor type="note" xlink:label="note-0243-03a" xlink:href="note-0243-03"/>
tem rectæ ET, GT curvarum duas (puta XEM, ZGO) in E, G; <lb/></s>
  <s xml:id="echoid-s11206" xml:space="preserve">oportet curvæ YFN tangentem ad F deſignare.</s>
  <s xml:id="echoid-s11207" xml:space="preserve"/>
</p>
<div xml:id="echoid-div309" type="float" level="2" n="9">
<note position="right" xlink:label="note-0243-03" xlink:href="note-0243-03a" xml:space="preserve">Fig. 82.</note>
</div>
<p>
  <s xml:id="echoid-s11208" xml:space="preserve">Concipiatur curva TFV talis, ut à D utcunque projectâ rectâ <lb/>DM K L, (quæ ſecet rectas TE, TG punctis I, L, &amp; </s>
  <s xml:id="echoid-s11209" xml:space="preserve">iſtam cur-<lb/>vam in K) habeant ſemper interceptæ IL, IK rationem eandem datæ <lb/>R ad S; </s>
  <s xml:id="echoid-s11210" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_a_)"/> eſt igitur IK &amp;</s>
  <s xml:id="echoid-s11211" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11212" xml:space="preserve">I N; </s>
  <s xml:id="echoid-s11213" xml:space="preserve">quare curva TFK curvam YFN
<anchor type="note" xlink:label="note-0243-04a" xlink:href="note-0243-04"/>
tangit; </s>
  <s xml:id="echoid-s11214" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_b_)"/> eſt antem curva TFK _hyperbola_; </s>
  <s xml:id="echoid-s11215" xml:space="preserve">hanc tangat FS; </s>
  <s xml:id="echoid-s11216" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_c_)"/>
<anchor type="note" xlink:label="note-0243-05a" xlink:href="note-0243-05"/>
<anchor type="note" xlink:label="note-0243-06a" xlink:href="note-0243-06"/>
illa quoque curvam YFN tanget.</s>
  <s xml:id="echoid-s11217" xml:space="preserve"/>
</p>
<div xml:id="echoid-div310" type="float" level="2" n="10">
<note position="right" xlink:label="note-0243-04" xlink:href="note-0243-04a" xml:space="preserve">(_a_) 2 Lect. <lb/>VIII.</note>
<note position="right" xlink:label="note-0243-05" xlink:href="note-0243-05a" xml:space="preserve">(_b_) 4 Lect. VI.</note>
<note position="right" xlink:label="note-0243-06" xlink:href="note-0243-06a" xml:space="preserve">(_c_) 2. _hujus<unsure/>_</note>
</div>
<p>
  <s xml:id="echoid-s11218" xml:space="preserve">Quoniam _hyperbolam_ tangentis hîc primum injecta eſt mentio; </s>
  <s xml:id="echoid-s11219" xml:space="preserve">hu-<lb/>jus ( unà cum aliarum omnium conſimili ratione procreatarum ſeu _re_-<lb/>_cipr ocarum linearum tangentibus_) _tangentem_ ità definiemus.</s>
  <s xml:id="echoid-s11220" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11221" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s11222" xml:space="preserve">Sint VD recta linea, duæque curvæ XEM, YFN ità re-<lb/>
<anchor type="note" xlink:label="note-0243-07a" xlink:href="note-0243-07"/>
latæ, ut ductâ liberè rectâ EDF ad poſitione datam parallelâ, ſit <lb/>ſemper _rectangulum_ ex DE, DF par eidem alicui ſpatio; </s>
  <s xml:id="echoid-s11223" xml:space="preserve">tangat au-<lb/>tem recta ET curvam XEM in E, cum recta VD concurrens in T; <lb/></s>
  <s xml:id="echoid-s11224" xml:space="preserve">ſumatúrque DS = DT; </s>
  <s xml:id="echoid-s11225" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11226" xml:space="preserve">connectatur F S; </s>
  <s xml:id="echoid-s11227" xml:space="preserve">hæc curvam YFN <lb/>tanget ad F.</s>
  <s xml:id="echoid-s11228" xml:space="preserve"/>
</p>
<div xml:id="echoid-div311" type="float" level="2" n="11">
<note position="right" xlink:label="note-0243-07" xlink:href="note-0243-07a" xml:space="preserve">Fig. 83.</note>
</div>
<p>
  <s xml:id="echoid-s11229" xml:space="preserve">Nam utcunque ducatur IN ad EF parallela; </s>
  <s xml:id="echoid-s11230" xml:space="preserve">lineas expoſitas ſe-
<pb o="66" file="0244" n="259" rhead=""/>
eans, ut vides. </s>
  <s xml:id="echoid-s11231" xml:space="preserve">Eſtque T P. </s>
  <s xml:id="echoid-s11232" xml:space="preserve">PM &amp;</s>
  <s xml:id="echoid-s11233" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11234" xml:space="preserve">( TP. </s>
  <s xml:id="echoid-s11235" xml:space="preserve">PI :</s>
  <s xml:id="echoid-s11236" xml:space="preserve">:) TD. </s>
  <s xml:id="echoid-s11237" xml:space="preserve">DE <lb/>item SP. </s>
  <s xml:id="echoid-s11238" xml:space="preserve">PK :</s>
  <s xml:id="echoid-s11239" xml:space="preserve">: SD. </s>
  <s xml:id="echoid-s11240" xml:space="preserve">DF. </s>
  <s xml:id="echoid-s11241" xml:space="preserve">ergò TP x SP. </s>
  <s xml:id="echoid-s11242" xml:space="preserve">PM x PK &amp;</s>
  <s xml:id="echoid-s11243" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11244" xml:space="preserve">TD <lb/>x SD. </s>
  <s xml:id="echoid-s11245" xml:space="preserve">DE xDF :</s>
  <s xml:id="echoid-s11246" xml:space="preserve">: TD x SD. </s>
  <s xml:id="echoid-s11247" xml:space="preserve">PM xPN. </s>
  <s xml:id="echoid-s11248" xml:space="preserve">Verum TD x <lb/>SD &amp;</s>
  <s xml:id="echoid-s11249" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11250" xml:space="preserve">TP xSP; </s>
  <s xml:id="echoid-s11251" xml:space="preserve">ac indè magís TD x SD. </s>
  <s xml:id="echoid-s11252" xml:space="preserve">PM x PK &amp;</s>
  <s xml:id="echoid-s11253" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11254" xml:space="preserve">TD x <lb/>SD. </s>
  <s xml:id="echoid-s11255" xml:space="preserve">PM x PN. </s>
  <s xml:id="echoid-s11256" xml:space="preserve">quare PM x PK &amp;</s>
  <s xml:id="echoid-s11257" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11258" xml:space="preserve">PM x PN; </s>
  <s xml:id="echoid-s11259" xml:space="preserve">vel PK &amp;</s>
  <s xml:id="echoid-s11260" xml:space="preserve">lt; <lb/></s>
  <s xml:id="echoid-s11261" xml:space="preserve">PN. </s>
  <s xml:id="echoid-s11262" xml:space="preserve">Itaque recta FS extra curvam YFN tota jacet.</s>
  <s xml:id="echoid-s11263" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11264" xml:space="preserve">_Not_. </s>
  <s xml:id="echoid-s11265" xml:space="preserve">Si linea XEM recta fuerit ( utique ipſi TE I coincidens) erit <lb/>YFN _hyperbola_ vulgaris, cujus centrum T, _aſymptotos_ una TS, al-<lb/>tera TZ ad EF parallela.</s>
  <s xml:id="echoid-s11266" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11267" xml:space="preserve">X. </s>
  <s xml:id="echoid-s11268" xml:space="preserve">Quinetiam ſit punctum D; </s>
  <s xml:id="echoid-s11269" xml:space="preserve">curvæque duæ XEM, YFN ità <lb/>relatæ, ut per D ductâ quacunque rectâ EF; </s>
  <s xml:id="echoid-s11270" xml:space="preserve">ſit perpetuo rectangu-<lb/>lum ex DE, DF æquale cuidam quadrato _(_ex Z puta); </s>
  <s xml:id="echoid-s11271" xml:space="preserve">unam verò <lb/>curvam XEM tangat recta ER; </s>
  <s xml:id="echoid-s11272" xml:space="preserve">alterius ad F tangens ita determina-<lb/>tur: </s>
  <s xml:id="echoid-s11273" xml:space="preserve">Ducatur DP ad ER perpendicularis: </s>
  <s xml:id="echoid-s11274" xml:space="preserve">factóque DP. </s>
  <s xml:id="echoid-s11275" xml:space="preserve">Z :</s>
  <s xml:id="echoid-s11276" xml:space="preserve">: Z. <lb/></s>
  <s xml:id="echoid-s11277" xml:space="preserve">
<anchor type="note" xlink:label="note-0244-01a" xlink:href="note-0244-01"/>
DB; </s>
  <s xml:id="echoid-s11278" xml:space="preserve">biſecetur DB in C; </s>
  <s xml:id="echoid-s11279" xml:space="preserve">connexâque CF, ducatur FS ad CF nor-<lb/>malis; </s>
  <s xml:id="echoid-s11280" xml:space="preserve">hæc curvam YFN tanget.</s>
  <s xml:id="echoid-s11281" xml:space="preserve"/>
</p>
<div xml:id="echoid-div312" type="float" level="2" n="12">
<note position="left" xlink:label="note-0244-01" xlink:href="note-0244-01a" xml:space="preserve">Fig. 84.</note>
</div>
<p>
  <s xml:id="echoid-s11282" xml:space="preserve">Nam centro C per F deſcribatur _Circulus_ DO B; </s>
  <s xml:id="echoid-s11283" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11284" xml:space="preserve">per B traji-<lb/>ciatur utcunque recta IN lineas interſecans, ut vides; </s>
  <s xml:id="echoid-s11285" xml:space="preserve">eſtque DO x <lb/>DI <anchor type="note" xlink:href="" symbol="(_a_)"/> = DP x DB <anchor type="note" xlink:href="" symbol="(_b_)"/> = Zq <anchor type="note" xlink:href="" symbol="(_c_)"/> = DM x DN vel DO. </s>
  <s xml:id="echoid-s11286" xml:space="preserve">DM
<anchor type="note" xlink:label="note-0244-02a" xlink:href="note-0244-02"/>
:</s>
  <s xml:id="echoid-s11287" xml:space="preserve">: DN. </s>
  <s xml:id="echoid-s11288" xml:space="preserve">DI. </s>
  <s xml:id="echoid-s11289" xml:space="preserve">ergò quum ſit DM (_c_) &amp;</s>
  <s xml:id="echoid-s11290" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11291" xml:space="preserve">DI; </s>
  <s xml:id="echoid-s11292" xml:space="preserve">erit DO &amp;</s>
  <s xml:id="echoid-s11293" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11294" xml:space="preserve">DN; <lb/></s>
  <s xml:id="echoid-s11295" xml:space="preserve">itaque circulus DOB curvam YFN tanget. </s>
  <s xml:id="echoid-s11296" xml:space="preserve">Quare recta FS eandem <lb/>
<anchor type="note" xlink:label="note-0244-03a" xlink:href="note-0244-03"/>
<anchor type="note" xlink:label="note-0244-04a" xlink:href="note-0244-04"/>
YF N tanget.</s>
  <s xml:id="echoid-s11297" xml:space="preserve"/>
</p>
<div xml:id="echoid-div313" type="float" level="2" n="13">
<note position="left" xlink:label="note-0244-02" xlink:href="note-0244-02a" xml:space="preserve">(_a_) 27 Lect. <lb/>VI.</note>
<note position="left" xlink:label="note-0244-03" xlink:href="note-0244-03a" xml:space="preserve">(_b_) _Conſtr_.</note>
<note position="left" xlink:label="note-0244-04" xlink:href="note-0244-04a" xml:space="preserve">(_c_)_Hyp._</note>
</div>
<p>
  <s xml:id="echoid-s11298" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s11299" xml:space="preserve">Curvæ XEM, YFN tales ſint, ut ductâ quâpiam FE ad poſi-<lb/>tione datam parallelâ, ſit ſemper hæc æqualis eidem alicui; </s>
  <s xml:id="echoid-s11300" xml:space="preserve">curvàm <lb/>autem YFN tangat recta SF; </s>
  <s xml:id="echoid-s11301" xml:space="preserve">huic parallela RE alteram XEM <lb/>
<anchor type="note" xlink:label="note-0244-05a" xlink:href="note-0244-05"/>
continget.</s>
  <s xml:id="echoid-s11302" xml:space="preserve"/>
</p>
<div xml:id="echoid-div314" type="float" level="2" n="14">
<note position="left" xlink:label="note-0244-05" xlink:href="note-0244-05a" xml:space="preserve">Fig. 85.</note>
</div>
<p>
  <s xml:id="echoid-s11303" xml:space="preserve">Nam utcunque ductâ MK ad FE parallelâ eſt NI &amp;</s>
  <s xml:id="echoid-s11304" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11305" xml:space="preserve">( KI = FE <lb/> = ) NM. </s>
  <s xml:id="echoid-s11306" xml:space="preserve">Quare punctum I extra curvam XEM jacet, _&amp; </s>
  <s xml:id="echoid-s11307" xml:space="preserve">c_.</s>
  <s xml:id="echoid-s11308" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11309" xml:space="preserve">Reverà linea XEM nil aliud eſt, quàm ipſa YFN _tranſlocata_. <lb/></s>
  <s xml:id="echoid-s11310" xml:space="preserve">Levius hoc, &amp; </s>
  <s xml:id="echoid-s11311" xml:space="preserve">methoditantùm gratiâ Propoſitum.</s>
  <s xml:id="echoid-s11312" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11313" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s11314" xml:space="preserve">Sit curva quæpiam XEM, quam tangat recta ER ad E; </s>
  <s xml:id="echoid-s11315" xml:space="preserve">ſit <lb/>item alia curva YF N ad alteram ità relata, ut ab aſſignato puncto D <lb/>
<anchor type="note" xlink:label="note-0244-06a" xlink:href="note-0244-06"/>
utcunque ductâ rectâ DEF, ſit ſemper intercepta EF æqualis alicui <lb/>determinatæ Z; </s>
  <s xml:id="echoid-s11316" xml:space="preserve">curvæ YFN tangens (ad F) ità deſignatur: </s>
  <s xml:id="echoid-s11317" xml:space="preserve">Su-<lb/>matur DH = Z; </s>
  <s xml:id="echoid-s11318" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11319" xml:space="preserve">per H ducatur AH ad DH perpendicularis, <lb/>ipſi ER occurrens in B, &amp; </s>
  <s xml:id="echoid-s11320" xml:space="preserve">per F ducatur FG ad AB parallela; </s>
  <s xml:id="echoid-s11321" xml:space="preserve">ſuma-<lb/>túrque GL = GB; </s>
  <s xml:id="echoid-s11322" xml:space="preserve">erit connexa LFS curvæ YFN tangens.</s>
  <s xml:id="echoid-s11323" xml:space="preserve"/>
</p>
<div xml:id="echoid-div315" type="float" level="2" n="15">
<note position="left" xlink:label="note-0244-06" xlink:href="note-0244-06a" xml:space="preserve">Fig. 86.</note>
</div>
<pb o="67" file="0245" n="260" rhead=""/>
<p>
  <s xml:id="echoid-s11324" xml:space="preserve">Nam _aſymptotis_ ER, AB per F deſcripta concipiatur _hyperbol@_ <lb/>OFO; </s>
  <s xml:id="echoid-s11325" xml:space="preserve">cui occurrat à D projecta quæpiam DO, lineas expoſitas <lb/>
<anchor type="note" xlink:label="note-0245-01a" xlink:href="note-0245-01"/>
ſecans, utì cernis. </s>
  <s xml:id="echoid-s11326" xml:space="preserve">Eſtque QO <anchor type="note" xlink:href="" symbol="(_a_)"/> = DP; </s>
  <s xml:id="echoid-s11327" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_b_)"/> quare MO &amp;</s>
  <s xml:id="echoid-s11328" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11329" xml:space="preserve">DP
<anchor type="note" xlink:label="note-0245-02a" xlink:href="note-0245-02"/>
<anchor type="note" xlink:label="note-0245-03a" xlink:href="note-0245-03"/>
<anchor type="note" xlink:href="" symbol="(_c_)"/> &amp;</s>
  <s xml:id="echoid-s11330" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11331" xml:space="preserve">DH <anchor type="note" xlink:href="" symbol="(_b_)"/> = MN. </s>
  <s xml:id="echoid-s11332" xml:space="preserve">ergò _hyperbola_ OFO curvam YFN tan-
git.</s>
  <s xml:id="echoid-s11333" xml:space="preserve"/>
</p>
<div xml:id="echoid-div316" type="float" level="2" n="16">
<note position="right" xlink:label="note-0245-01" xlink:href="note-0245-01a" xml:space="preserve">(_a_) Conver ſ 9@ <lb/>Lect. VI.</note>
<note position="right" xlink:label="note-0245-02" xlink:href="note-0245-02a" xml:space="preserve">(_b_)_Hyp_.</note>
<note position="right" xlink:label="note-0245-03" xlink:href="note-0245-03a" xml:space="preserve">(_c_)_Elem_.</note>
</div>
<p>
  <s xml:id="echoid-s11334" xml:space="preserve">Verùm <anchor type="note" xlink:href="" symbol="(_d_)"/> recta LS _hyperbolam_ OFO tangit; </s>
  <s xml:id="echoid-s11335" xml:space="preserve">hæc itaque curvam
<anchor type="note" xlink:label="note-0245-04a" xlink:href="note-0245-04"/>
YF N quoque tanget.</s>
  <s xml:id="echoid-s11336" xml:space="preserve"/>
</p>
<div xml:id="echoid-div317" type="float" level="2" n="17">
<note position="right" xlink:label="note-0245-04" xlink:href="note-0245-04a" xml:space="preserve">(_d_)_9. hujues_</note>
</div>
<p>
  <s xml:id="echoid-s11337" xml:space="preserve">_Not_. </s>
  <s xml:id="echoid-s11338" xml:space="preserve">Si XEM ponatur linea recta ( vel ipſi ER coincidat) erit <lb/>YF N _Conchois_ prima vulgaris, ſeu _Nicomedea_; </s>
  <s xml:id="echoid-s11339" xml:space="preserve">hujus igitur tangens <lb/>è generali ratione quâdam habetur determinata.</s>
  <s xml:id="echoid-s11340" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11341" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s11342" xml:space="preserve">Sit recta LA, curváque quæpiam BEI; </s>
  <s xml:id="echoid-s11343" xml:space="preserve">cum alia curva <lb/>DFG talis, ut ductâ liberè rectâ PFE ad p@itione datâm quandam <lb/>
<anchor type="note" xlink:label="note-0245-05a" xlink:href="note-0245-05"/>
parallelâ, poſſit recta PE quadratum ex PF unà cum quadrato ex da-<lb/>tâ Z; </s>
  <s xml:id="echoid-s11344" xml:space="preserve">item curvam BE I tangat recta ET; </s>
  <s xml:id="echoid-s11345" xml:space="preserve">tum fiat PEq. </s>
  <s xml:id="echoid-s11346" xml:space="preserve">PFq :</s>
  <s xml:id="echoid-s11347" xml:space="preserve">: <lb/>PT. </s>
  <s xml:id="echoid-s11348" xml:space="preserve">PS; </s>
  <s xml:id="echoid-s11349" xml:space="preserve">connexa SF curvam DFG tanget.</s>
  <s xml:id="echoid-s11350" xml:space="preserve"/>
</p>
<div xml:id="echoid-div318" type="float" level="2" n="18">
<note position="right" xlink:label="note-0245-05" xlink:href="note-0245-05a" xml:space="preserve">Fig. 87.</note>
</div>
<p>
  <s xml:id="echoid-s11351" xml:space="preserve">Nam concipiatur curva VFH talis, ut liberè ductâ QK ad PE <lb/>parallelâ (quæ lineas expoſitas ſecet ut vides) ſit perpetuò QKq = <lb/>QHq + Zq; </s>
  <s xml:id="echoid-s11352" xml:space="preserve">unde quoniam eſt QK <anchor type="note" xlink:href="" symbol="(_a_)"/> &amp;</s>
  <s xml:id="echoid-s11353" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11354" xml:space="preserve">Q I; </s>
  <s xml:id="echoid-s11355" xml:space="preserve">erit QKq --
<anchor type="note" xlink:label="note-0245-06a" xlink:href="note-0245-06"/>
Zq &amp;</s>
  <s xml:id="echoid-s11356" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11357" xml:space="preserve">Q Iq-- Zq; </s>
  <s xml:id="echoid-s11358" xml:space="preserve">hoc eſt QHq &amp;</s>
  <s xml:id="echoid-s11359" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11360" xml:space="preserve">QGq; </s>
  <s xml:id="echoid-s11361" xml:space="preserve">ergò curva VFH <lb/>
<anchor type="note" xlink:label="note-0245-07a" xlink:href="note-0245-07"/>
curvam DFG tanget ad F ; </s>
  <s xml:id="echoid-s11362" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_b_)"/>eſt autem curva VF H _hyperbola_, quam
<anchor type="note" xlink:label="note-0245-08a" xlink:href="note-0245-08"/>
<anchor type="note" xlink:href="" symbol="(_c_)"/> tangit recta SF. </s>
  <s xml:id="echoid-s11363" xml:space="preserve">hæc itaque curvam DFG quoque contin-
get.</s>
  <s xml:id="echoid-s11364" xml:space="preserve"/>
</p>
<div xml:id="echoid-div319" type="float" level="2" n="19">
<note position="right" xlink:label="note-0245-06" xlink:href="note-0245-06a" xml:space="preserve">(_a_)_Hyp_.</note>
<note position="right" xlink:label="note-0245-07" xlink:href="note-0245-07a" xml:space="preserve">(_b_) 22. _Lect. 6._</note>
<note position="right" xlink:label="note-0245-08" xlink:href="note-0245-08a" xml:space="preserve">(_c_)_Cor. 22._ <lb/>Lect. _I_.</note>
</div>
<p>
  <s xml:id="echoid-s11365" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s11366" xml:space="preserve">Cætera ponantur eadem; </s>
  <s xml:id="echoid-s11367" xml:space="preserve">at jam PE unà cum quadrato ex <lb/>data Z poſſit quadratum ex PF; </s>
  <s xml:id="echoid-s11368" xml:space="preserve">fiátque PEq. </s>
  <s xml:id="echoid-s11369" xml:space="preserve">PFq :</s>
  <s xml:id="echoid-s11370" xml:space="preserve">: PT. </s>
  <s xml:id="echoid-s11371" xml:space="preserve">PS; <lb/></s>
  <s xml:id="echoid-s11372" xml:space="preserve">
<anchor type="note" xlink:label="note-0245-09a" xlink:href="note-0245-09"/>
&amp; </s>
  <s xml:id="echoid-s11373" xml:space="preserve">connectatur FS; </s>
  <s xml:id="echoid-s11374" xml:space="preserve">hæc rurſus ipſam GFG continget.</s>
  <s xml:id="echoid-s11375" xml:space="preserve"/>
</p>
<div xml:id="echoid-div320" type="float" level="2" n="20">
<note position="right" xlink:label="note-0245-09" xlink:href="note-0245-09a" xml:space="preserve">Fig. 88.</note>
</div>
<p>
  <s xml:id="echoid-s11376" xml:space="preserve">Similis eſt demonſtratio; </s>
  <s xml:id="echoid-s11377" xml:space="preserve">ſed adhibe 23 am primæ Lectionis.</s>
  <s xml:id="echoid-s11378" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11379" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s11380" xml:space="preserve">Sint curvæ duæ AFB, CGD, communem habentes _axe@@_ <lb/>AD, ac ità verſus ſerelatæ, ut ductâ quâcunque rectâ FEG ad AD <lb/>perpendiculari ( quæ rectas expoſitas ſecet ut vides ) ſit ſumma qua-<lb/>dratorum ex ipſis EF, EG æqualis quadrato ex determinata recta Z; <lb/></s>
  <s xml:id="echoid-s11381" xml:space="preserve">
<anchor type="note" xlink:label="note-0245-10a" xlink:href="note-0245-10"/>
tangat autem recta FR ex his curvis unam AFB; </s>
  <s xml:id="echoid-s11382" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11383" xml:space="preserve">fiat EFq. <lb/></s>
  <s xml:id="echoid-s11384" xml:space="preserve">EGq :</s>
  <s xml:id="echoid-s11385" xml:space="preserve">: ER. </s>
  <s xml:id="echoid-s11386" xml:space="preserve">ET; </s>
  <s xml:id="echoid-s11387" xml:space="preserve">connexa GT curvam CGD quoque tanget.</s>
  <s xml:id="echoid-s11388" xml:space="preserve"/>
</p>
<div xml:id="echoid-div321" type="float" level="2" n="21">
<note position="right" xlink:label="note-0245-10" xlink:href="note-0245-10a" xml:space="preserve">Fig. 89.</note>
</div>
<p>
  <s xml:id="echoid-s11389" xml:space="preserve">Concipiatur enim curva OGO talis, ut ductâ rectâ KQO (quæ <lb/>rectas FR, ER ſecet punctis K, Q, curvam OGO in O ) ſit QKq <lb/>+ QO = Zq; </s>
  <s xml:id="echoid-s11390" xml:space="preserve">erit ideò QKq + QOq = QIq + QLq; <lb/></s>
  <s xml:id="echoid-s11391" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11392" xml:space="preserve">cùm ſit QKq <anchor type="note" xlink:href="" symbol="(_a_)"/> &amp;</s>
  <s xml:id="echoid-s11393" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11394" xml:space="preserve">QIq, erit ideò QOq &amp;</s>
  <s xml:id="echoid-s11395" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11396" xml:space="preserve">QLq. </s>
  <s xml:id="echoid-s11397" xml:space="preserve">itaque
<anchor type="note" xlink:label="note-0245-11a" xlink:href="note-0245-11"/>
curva OGO curvam CGD (introrſum) tangit. </s>
  <s xml:id="echoid-s11398" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_b_)"/> Eſt autem ( ex
<anchor type="note" xlink:label="note-0245-12a" xlink:href="note-0245-12"/>
<pb o="68" file="0246" n="261" rhead=""/>
oſtenſis) curva OGO _Ellipſis_, quam recta GT tangit. </s>
  <s xml:id="echoid-s11399" xml:space="preserve">ergò recta <lb/>GT curvam CGD quoque tanget.</s>
  <s xml:id="echoid-s11400" xml:space="preserve"/>
</p>
<div xml:id="echoid-div322" type="float" level="2" n="22">
<note position="right" xlink:label="note-0245-11" xlink:href="note-0245-11a" xml:space="preserve">(_a_)_Hyp_.</note>
<note position="right" xlink:label="note-0245-12" xlink:href="note-0245-12a" xml:space="preserve">(_b_) 24. lect. <lb/>VI.</note>
</div>
<p>
  <s xml:id="echoid-s11401" xml:space="preserve">XVI Sit curva quæpiam AFB (cujus axis AD, &amp; </s>
  <s xml:id="echoid-s11402" xml:space="preserve">ad hunc ap-<lb/>
<anchor type="note" xlink:label="note-0246-01a" xlink:href="note-0246-01"/>
plicata DB) ſit etiam alia curva VGC ad iſtam ſic relata, ut a deſig-<lb/>nato quodam in axe AD puncto Z ad curvam AFB utcunque ductâ <lb/>rectâ ZF, &amp; </s>
  <s xml:id="echoid-s11403" xml:space="preserve">per F ductâ rectâ EFG ad DBC parallelâ, ſit EG <lb/>æqualis ipſi ZF; </s>
  <s xml:id="echoid-s11404" xml:space="preserve">ſit autem PQ perpendicularis curvæ AFB; </s>
  <s xml:id="echoid-s11405" xml:space="preserve">ſu-<lb/>matúruqe QR æqualis ipſi ZE; </s>
  <s xml:id="echoid-s11406" xml:space="preserve">connexa recta GR ipſi curvæ VGC <lb/>perpendicularis erit.</s>
  <s xml:id="echoid-s11407" xml:space="preserve"/>
</p>
<div xml:id="echoid-div323" type="float" level="2" n="23">
<note position="left" xlink:label="note-0246-01" xlink:href="note-0246-01a" xml:space="preserve">Fig. 90.</note>
</div>
<p>
  <s xml:id="echoid-s11408" xml:space="preserve">Nam ducatur FT ad ipſam FQ perpendicularis, ſeu curvam AFB <lb/>tangens; </s>
  <s xml:id="echoid-s11409" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11410" xml:space="preserve">concipiatur curva OGO talis, ut ductâ quâcunq; </s>
  <s xml:id="echoid-s11411" xml:space="preserve">rectâ <lb/>HKO ad EFG parallelâ ( quæ rectas TE, TF, &amp; </s>
  <s xml:id="echoid-s11412" xml:space="preserve">curvam OGO <lb/>ſecet punctis H, K, O) connexâque ZK, ſit HO = ZK; </s>
  <s xml:id="echoid-s11413" xml:space="preserve">tum du-<lb/>
<anchor type="note" xlink:label="note-0246-02a" xlink:href="note-0246-02"/>
ctâ Z I, quoniam HK <anchor type="note" xlink:href="" symbol="(_a_)"/> &amp;</s>
  <s xml:id="echoid-s11414" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11415" xml:space="preserve">HI, erit ZK &amp;</s>
  <s xml:id="echoid-s11416" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11417" xml:space="preserve">ZI, vel HO &amp;</s>
  <s xml:id="echoid-s11418" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11419" xml:space="preserve">HL;</s>
  <s xml:id="echoid-s11420" xml:space="preserve">
<anchor type="note" xlink:label="note-0246-03a" xlink:href="note-0246-03"/>
quare curva OGO curvam VGC tangit. </s>
  <s xml:id="echoid-s11421" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_b_)"/> Eſt autem OGO
(ex oſtenſis) _Hyperbola_, cui perpendicularis eſt recta GR; </s>
  <s xml:id="echoid-s11422" xml:space="preserve">eadem <lb/>itaque GR curvæ VGC quoque perpendicularis erit: </s>
  <s xml:id="echoid-s11423" xml:space="preserve">Quod E. </s>
  <s xml:id="echoid-s11424" xml:space="preserve">D.</s>
  <s xml:id="echoid-s11425" xml:space="preserve"/>
</p>
<div xml:id="echoid-div324" type="float" level="2" n="24">
<note position="left" xlink:label="note-0246-02" xlink:href="note-0246-02a" xml:space="preserve">(_a_)_Hyp_.</note>
<note position="left" xlink:label="note-0246-03" xlink:href="note-0246-03a" xml:space="preserve">(_b_)25 Lect. VI</note>
</div>
<p>
  <s xml:id="echoid-s11426" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s11427" xml:space="preserve">Sint recta DQ, duæque curvæ DRS, DYX ità relatæ, <lb/>
<anchor type="note" xlink:label="note-0246-04a" xlink:href="note-0246-04"/>
ut ductâ utcunque rectâ REY ad poſitione datam DB parallelâ (quæ <lb/>dictas lineas ſecet, ut perſpicis) connexâque rectâ DY, ſit ſemper <lb/>RY. </s>
  <s xml:id="echoid-s11428" xml:space="preserve">DY :</s>
  <s xml:id="echoid-s11429" xml:space="preserve">: DY. </s>
  <s xml:id="echoid-s11430" xml:space="preserve">EY; </s>
  <s xml:id="echoid-s11431" xml:space="preserve">tangat autem recta RF curvam DRS ad R; <lb/></s>
  <s xml:id="echoid-s11432" xml:space="preserve">oporter curvæ DYX tangentem ad Y rectam deſignare.</s>
  <s xml:id="echoid-s11433" xml:space="preserve"/>
</p>
<div xml:id="echoid-div325" type="float" level="2" n="25">
<note position="left" xlink:label="note-0246-04" xlink:href="note-0246-04a" xml:space="preserve">Fig. 91.</note>
</div>
<p>
  <s xml:id="echoid-s11434" xml:space="preserve">Concipiatur linea DYO talis, ut ductâ utcunque GO ad DB pa-<lb/>rallelâ ( quæ lineas FR, FP, DYO ſecet punctis G, P, O) connexâ-<lb/>que DO ſit ſemper GO. </s>
  <s xml:id="echoid-s11435" xml:space="preserve">DO_:_</s>
  <s xml:id="echoid-s11436" xml:space="preserve">: </s>
  <s xml:id="echoid-s11437" xml:space="preserve">DO. </s>
  <s xml:id="echoid-s11438" xml:space="preserve">PO; </s>
  <s xml:id="echoid-s11439" xml:space="preserve">tanget curva DYO <lb/>curvam DYX ad Y; </s>
  <s xml:id="echoid-s11440" xml:space="preserve">Nam ſecet recta GO curvas DRS, DYX <lb/>punctis S, X; </s>
  <s xml:id="echoid-s11441" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11442" xml:space="preserve">connectantur rectæ DG, DS, DX; </s>
  <s xml:id="echoid-s11443" xml:space="preserve">patet (è cur-<lb/>varum natura) angulos XDP, DSP; </s>
  <s xml:id="echoid-s11444" xml:space="preserve">nec non angulos ODP, DGP <lb/>æquari; </s>
  <s xml:id="echoid-s11445" xml:space="preserve">quare cùm angulus DSP major ſit angulo DGP; </s>
  <s xml:id="echoid-s11446" xml:space="preserve">erit an-<lb/>gulus XDP angulo ODP major, adeóque PX major erit quàm PO; <lb/></s>
  <s xml:id="echoid-s11447" xml:space="preserve">
<anchor type="note" xlink:label="note-0246-05a" xlink:href="note-0246-05"/>
hinc curva DYO curvam DYX tanget ad Y; </s>
  <s xml:id="echoid-s11448" xml:space="preserve">eſt autem curva DYO <lb/>_hyperbolæ_ <anchor type="note" xlink:href="" symbol="(_a_)"/> ſuperiùs determinata; </s>
  <s xml:id="echoid-s11449" xml:space="preserve">hanc tangat YS; </s>
  <s xml:id="echoid-s11450" xml:space="preserve">hæc igitur
curvam DYX quoque tanget.</s>
  <s xml:id="echoid-s11451" xml:space="preserve"/>
</p>
<div xml:id="echoid-div326" type="float" level="2" n="26">
<note position="left" xlink:label="note-0246-05" xlink:href="note-0246-05a" xml:space="preserve">(_a_) Y2 Lect. <lb/>VI.</note>
</div>
<p>
  <s xml:id="echoid-s11452" xml:space="preserve">_Not_. </s>
  <s xml:id="echoid-s11453" xml:space="preserve">Si curva DRS ſit circulus, &amp; </s>
  <s xml:id="echoid-s11454" xml:space="preserve">angulus QDB rectus, erit cur-<lb/>va DYX _ciſſois_ vulgaris; </s>
  <s xml:id="echoid-s11455" xml:space="preserve">hujus itaque ( cum innumeris aliis ſimiliter <lb/>genitis) tangens hîc deſinitur.</s>
  <s xml:id="echoid-s11456" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11457" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s11458" xml:space="preserve">Poſitione datæ ſint rectæ DB, BK; </s>
  <s xml:id="echoid-s11459" xml:space="preserve">ſitque curva DYX <lb/>
<anchor type="note" xlink:label="note-0246-06a" xlink:href="note-0246-06"/>
<pb o="69" file="0247" n="262" rhead=""/>
talis; </s>
  <s xml:id="echoid-s11460" xml:space="preserve">ut à puncto D ductâ quâvis rectâ DYH (quæ rectam BK ſe-<lb/>cet in H, curvam DYX in Y) ſit perpetuò ſubtenſa DY æqualis re-<lb/>ctæ BH; </s>
  <s xml:id="echoid-s11461" xml:space="preserve">oportet curvæ DYX tangentem ad Y rectam determi-<lb/>nare.</s>
  <s xml:id="echoid-s11462" xml:space="preserve"/>
</p>
<div xml:id="echoid-div327" type="float" level="2" n="27">
<note position="left" xlink:label="note-0246-06" xlink:href="note-0246-06a" xml:space="preserve">Fig. 92.</note>
</div>
<p>
  <s xml:id="echoid-s11463" xml:space="preserve">Centro D per B ducatur circulus BRS; </s>
  <s xml:id="echoid-s11464" xml:space="preserve">cui occurrat recta YER <lb/>ad BK parallela; </s>
  <s xml:id="echoid-s11465" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11466" xml:space="preserve">connectatur DR; </s>
  <s xml:id="echoid-s11467" xml:space="preserve">eſtque (propter ang. </s>
  <s xml:id="echoid-s11468" xml:space="preserve">DYE <lb/> = ang. </s>
  <s xml:id="echoid-s11469" xml:space="preserve">DHB; </s>
  <s xml:id="echoid-s11470" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11471" xml:space="preserve">DY = BH, ac DR = DB) triangulum RDY <lb/>triangulo DBH ſimile ac æquale; </s>
  <s xml:id="echoid-s11472" xml:space="preserve">quare RY. </s>
  <s xml:id="echoid-s11473" xml:space="preserve">YD :</s>
  <s xml:id="echoid-s11474" xml:space="preserve">: (DH. </s>
  <s xml:id="echoid-s11475" xml:space="preserve">HB) <lb/>:</s>
  <s xml:id="echoid-s11476" xml:space="preserve">: YD. </s>
  <s xml:id="echoid-s11477" xml:space="preserve">YE. </s>
  <s xml:id="echoid-s11478" xml:space="preserve">unde ex præcedente determinabilis eſt recta curvam <lb/>DYX tangens in Y.</s>
  <s xml:id="echoid-s11479" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11480" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s11481" xml:space="preserve">Sint itidem rectæ DB, BK poſitione datæ; </s>
  <s xml:id="echoid-s11482" xml:space="preserve">nec non curva <lb/>BXX talis, ut à puncto Dprojectâ quâcunque rectâ DX (quæ re-<lb/>
<anchor type="note" xlink:label="note-0247-01a" xlink:href="note-0247-01"/>
ctam BK ſecet in H, curvámque BXX in X) ſit perpetuò HX ipſi <lb/>BH æqualis; </s>
  <s xml:id="echoid-s11483" xml:space="preserve">deſignetur oportet recta curvam BMX tangens in X.</s>
  <s xml:id="echoid-s11484" xml:space="preserve"/>
</p>
<div xml:id="echoid-div328" type="float" level="2" n="28">
<note position="right" xlink:label="note-0247-01" xlink:href="note-0247-01a" xml:space="preserve">Fig. 93.</note>
</div>
<p>
  <s xml:id="echoid-s11485" xml:space="preserve">Concipiatur curva DYY talis, ut perpetuò ſit DY = BH (talis <lb/>nempe, qualem attigimus in præcedente) hanc verò tangat recta YT <lb/>in Y, ipſi BK occurrens in R; </s>
  <s xml:id="echoid-s11486" xml:space="preserve">tum _aſymptotis_ RB, RT per X de-<lb/>ſcripta cenſeatur _hyperbola_ NXN; </s>
  <s xml:id="echoid-s11487" xml:space="preserve">ad quam utcunque projiciatur re-<lb/>
<anchor type="note" xlink:label="note-0247-02a" xlink:href="note-0247-02"/>
cta DN (lineas expoſitas ſecans, ut vides) Eſtque jam OM <anchor type="note" xlink:href="" symbol="(_a_)"/> =
<anchor type="note" xlink:label="note-0247-03a" xlink:href="note-0247-03"/>
D I) &amp;</s>
  <s xml:id="echoid-s11488" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11489" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_a_)"/> (DL <anchor type="note" xlink:href="" symbol="_(b)_"/> = ) ON; </s>
  <s xml:id="echoid-s11490" xml:space="preserve">ergò _hyperbola_ NXN curvam BXX
tangit ad X. </s>
  <s xml:id="echoid-s11491" xml:space="preserve">Ducatur itaque recta XS _hyperbolam_ NXN contin-<lb/>gens, hæc ipſam curvam BXX quoque continget.</s>
  <s xml:id="echoid-s11492" xml:space="preserve"/>
</p>
<div xml:id="echoid-div329" type="float" level="2" n="29">
<note position="right" xlink:label="note-0247-02" xlink:href="note-0247-02a" xml:space="preserve">_a_) _Couſtr_.</note>
<note position="right" xlink:label="note-0247-03" xlink:href="note-0247-03a" xml:space="preserve">(_b_) _Converſ_ <lb/>9. Lect. VI</note>
</div>
<p>
  <s xml:id="echoid-s11493" xml:space="preserve">Cæterùm ſatìs pro hac vice nugati videmur; </s>
  <s xml:id="echoid-s11494" xml:space="preserve">ceſſemus aliquantiſper.</s>
  <s xml:id="echoid-s11495" xml:space="preserve"/>
</p>
<pb o="70" file="0248" n="263"/>
</div>
<div xml:id="echoid-div331" type="section" level="1" n="34">
<head xml:id="echoid-head37" xml:space="preserve"><emph style="sc">Lect</emph>. IX.</head>
<p>
  <s xml:id="echoid-s11496" xml:space="preserve">QUod ingreſſi ſumus iter actutùm rectà proſequemur.</s>
  <s xml:id="echoid-s11497" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11498" xml:space="preserve">1. </s>
  <s xml:id="echoid-s11499" xml:space="preserve">Sint rectæ AB, VD ſibi parallelæ; </s>
  <s xml:id="echoid-s11500" xml:space="preserve">quas ſecat poſitione data <lb/>DB; </s>
  <s xml:id="echoid-s11501" xml:space="preserve">tranſeant verò per B lineæ EBE, FBF ità ad ſe relatæ, ut <lb/>
<anchor type="note" xlink:label="note-0248-01a" xlink:href="note-0248-01"/>
ductâ quâvis PG ad DB parallelâ, ſit perpetuò PF inter PG, PE <lb/>eodem ordine deſignato media _Arithmeticè_; </s>
  <s xml:id="echoid-s11502" xml:space="preserve">tangat autem recta BS <lb/>curvam EBE; </s>
  <s xml:id="echoid-s11503" xml:space="preserve">oportet lineæ FB F tangentem (ad B) deſignare.</s>
  <s xml:id="echoid-s11504" xml:space="preserve"/>
</p>
<div xml:id="echoid-div331" type="float" level="2" n="1">
<note position="left" xlink:label="note-0248-01" xlink:href="note-0248-01a" xml:space="preserve">Fig. 94.</note>
</div>
<p>
  <s xml:id="echoid-s11505" xml:space="preserve">Sint Numeri N, M proportionalium PF, PE (quales <anchor type="note" xlink:href="" symbol="(_a_)"/> explicu-
<anchor type="note" xlink:label="note-0248-02a" xlink:href="note-0248-02"/>
imus ſupra) exponentes; </s>
  <s xml:id="echoid-s11506" xml:space="preserve">fiátque N. </s>
  <s xml:id="echoid-s11507" xml:space="preserve">M :</s>
  <s xml:id="echoid-s11508" xml:space="preserve">: DS. </s>
  <s xml:id="echoid-s11509" xml:space="preserve">DT; </s>
  <s xml:id="echoid-s11510" xml:space="preserve">connectatúr-<lb/>que TB; </s>
  <s xml:id="echoid-s11511" xml:space="preserve">hæc lineam FBF continget.</s>
  <s xml:id="echoid-s11512" xml:space="preserve"/>
</p>
<div xml:id="echoid-div332" type="float" level="2" n="2">
<note position="left" xlink:label="note-0248-02" xlink:href="note-0248-02a" xml:space="preserve">(_a_) 12. Lect. <lb/>VII.</note>
</div>
<p>
  <s xml:id="echoid-s11513" xml:space="preserve">Nam utcúnque ducta ſit recta PG, dictas lineas ſecans, utì cernis: <lb/></s>
  <s xml:id="echoid-s11514" xml:space="preserve">Eſtque FG. </s>
  <s xml:id="echoid-s11515" xml:space="preserve">EG <anchor type="note" xlink:href="" symbol="(_b_)"/> :</s>
  <s xml:id="echoid-s11516" xml:space="preserve">: N. </s>
  <s xml:id="echoid-s11517" xml:space="preserve">M :</s>
  <s xml:id="echoid-s11518" xml:space="preserve">: <anchor type="note" xlink:href="" symbol="(_c_)"/> DS. </s>
  <s xml:id="echoid-s11519" xml:space="preserve">DT :</s>
  <s xml:id="echoid-s11520" xml:space="preserve">: <anchor type="note" xlink:href="" symbol="(_d_)"/> LG. </s>
  <s xml:id="echoid-s11521" xml:space="preserve">KG;</s>
  <s xml:id="echoid-s11522" xml:space="preserve">
<anchor type="note" xlink:label="note-0248-03a" xlink:href="note-0248-03"/>
cùm ergò <anchor type="note" xlink:href="" symbol="(_c_)"/> ſit KG &amp;</s>
  <s xml:id="echoid-s11523" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11524" xml:space="preserve">EG; </s>
  <s xml:id="echoid-s11525" xml:space="preserve">erit LG &amp;</s>
  <s xml:id="echoid-s11526" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11527" xml:space="preserve">FG; </s>
  <s xml:id="echoid-s11528" xml:space="preserve">unde liquet rectam
<anchor type="note" xlink:label="note-0248-04a" xlink:href="note-0248-04"/>
TB extra curvam FBF totam conſiſtere.</s>
  <s xml:id="echoid-s11529" xml:space="preserve"/>
</p>
<div xml:id="echoid-div333" type="float" level="2" n="3">
<note position="left" xlink:label="note-0248-03" xlink:href="note-0248-03a" xml:space="preserve">(_b_@@ Lect. <lb/>VII.</note>
<note position="left" xlink:label="note-0248-04" xlink:href="note-0248-04a" xml:space="preserve">(_c_) _Conſtr_.</note>
</div>
<note position="left" xml:space="preserve">(_d_) 3. Lect. 7.</note>
<note position="left" xml:space="preserve">(_e_) _Hyp_.</note>
<p>
  <s xml:id="echoid-s11530" xml:space="preserve">II. </s>
  <s xml:id="echoid-s11531" xml:space="preserve">Reliquis perſtantibus iiſdem, ſit jam PF inter PG, PE media pro-<lb/>portionalis Geometrìcè (eodem ordine media nempe, quo fuit priùs <lb/>Arithmeticè) eadem BT curvam FB F continget.</s>
  <s xml:id="echoid-s11532" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11533" xml:space="preserve">Etenim è mediis Arithmeticè Geometricéque proportionalibus hoc-<lb/>
<anchor type="note" xlink:label="note-0248-07a" xlink:href="note-0248-07"/>
ce modo conſtructæ lineæ ſeſe mutuo <anchor type="note" xlink:href="" symbol="(_a_)"/> contingunt ad B; </s>
  <s xml:id="echoid-s11534" xml:space="preserve">ergò cùm
recta BT tangat unam, hæc alteram quoque continget.</s>
  <s xml:id="echoid-s11535" xml:space="preserve"/>
</p>
<div xml:id="echoid-div334" type="float" level="2" n="4">
<note position="left" xlink:label="note-0248-07" xlink:href="note-0248-07a" xml:space="preserve">(_a_) 17. Lect. <lb/>7.</note>
</div>
<p>
  <s xml:id="echoid-s11536" xml:space="preserve">_Exemplum_. </s>
  <s xml:id="echoid-s11537" xml:space="preserve">Sit PF inter PG, PE è fex mediis tertia; </s>
  <s xml:id="echoid-s11538" xml:space="preserve">erit er-<lb/>go M = 7; </s>
  <s xml:id="echoid-s11539" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11540" xml:space="preserve">N = 3; </s>
  <s xml:id="echoid-s11541" xml:space="preserve">adeóque DS. </s>
  <s xml:id="echoid-s11542" xml:space="preserve">DT :</s>
  <s xml:id="echoid-s11543" xml:space="preserve">: 3.</s>
  <s xml:id="echoid-s11544" xml:space="preserve">7.</s>
  <s xml:id="echoid-s11545" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11546" xml:space="preserve">III. </s>
  <s xml:id="echoid-s11547" xml:space="preserve">Manente porrò quoad cætera proximè præcedente hypotheſi, <lb/>
<anchor type="note" xlink:label="note-0248-08a" xlink:href="note-0248-08"/>
ſumptóque quovis in curva FBF puncto F; </s>
  <s xml:id="echoid-s11548" xml:space="preserve">etiam ad hoc punctum <lb/>tangens recta ſimili pacto deſignatur.</s>
  <s xml:id="echoid-s11549" xml:space="preserve"/>
</p>
<div xml:id="echoid-div335" type="float" level="2" n="5">
<note position="left" xlink:label="note-0248-08" xlink:href="note-0248-08a" xml:space="preserve">Fig. 95.</note>
</div>
<p>
  <s xml:id="echoid-s11550" xml:space="preserve">Nempe per F ducatur recta PG ad ipſam DB parallela, ſecans <lb/>curvam EXE in E, tum EX tangat curvam EBE in E; </s>
  <s xml:id="echoid-s11551" xml:space="preserve">fiátque N.</s>
  <s xml:id="echoid-s11552" xml:space="preserve">
<pb o="71" file="0249" n="264" rhead=""/>
M :</s>
  <s xml:id="echoid-s11553" xml:space="preserve">: PX. </s>
  <s xml:id="echoid-s11554" xml:space="preserve">PY; </s>
  <s xml:id="echoid-s11555" xml:space="preserve">connectatúrque recta FY; </s>
  <s xml:id="echoid-s11556" xml:space="preserve">hæc curvam FBF continget.</s>
  <s xml:id="echoid-s11557" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11558" xml:space="preserve">Nam per E ducatur recta CE ad AB (vel VD) parallela; </s>
  <s xml:id="echoid-s11559" xml:space="preserve">conci-<lb/>piatúrque @@@ E tranſiens curva HEH talis, ut ductâ quâpiam QL <lb/>ad DE parallelâ (curvas EBE, HEH in L, &amp; </s>
  <s xml:id="echoid-s11560" xml:space="preserve">H; </s>
  <s xml:id="echoid-s11561" xml:space="preserve">rectáſque CE, <lb/>VP in I ac Q ſecante ) ſit ſemper QH inter QI, QL eodem ordine <lb/>media, quo PF inter PG, PE; </s>
  <s xml:id="echoid-s11562" xml:space="preserve">è præcedente jam conſtat rectam <lb/>connexam EY curvam HEH contingere; </s>
  <s xml:id="echoid-s11563" xml:space="preserve">verùm curvæ HEH <anchor type="note" xlink:href="" symbol="(_a_)"/>
analoga eſt curva FBF; </s>
  <s xml:id="echoid-s11564" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_b_)"/> ergò recta FY curvam FBF quoque
<anchor type="note" xlink:label="note-0249-01a" xlink:href="note-0249-01"/>
continget.</s>
  <s xml:id="echoid-s11565" xml:space="preserve"/>
</p>
<div xml:id="echoid-div336" type="float" level="2" n="6">
<note position="right" xlink:label="note-0249-01" xlink:href="note-0249-01a" xml:space="preserve">_a_ 7. Lect. 7.</note>
</div>
<note position="right" xml:space="preserve">_b_ 5. Lect. 8.</note>
<p>
  <s xml:id="echoid-s11566" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s11567" xml:space="preserve">Adnotetur, poſito lineam EBE rectam eſſe, quòd linea FBF <lb/>parabolarum ſeu paraboliſormium aliqua ſit. </s>
  <s xml:id="echoid-s11568" xml:space="preserve">quare quod de his paſ-<lb/>ſim obſervatum habetur _(_ex calculo deduc@um, &amp; </s>
  <s xml:id="echoid-s11569" xml:space="preserve">inductione quâdam <lb/>comprobatum, neſcio tamen an uſpiam Geometricè oſtenſum ) ex im-<lb/>mensùm uberiore fonte manat, ad iunumeras aliorum generum curvas <lb/>ſe diffundente.</s>
  <s xml:id="echoid-s11570" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11571" xml:space="preserve">V. </s>
  <s xml:id="echoid-s11572" xml:space="preserve">Hinc apertè conſectatur; </s>
  <s xml:id="echoid-s11573" xml:space="preserve">ſi TD ſit recta, síntque duæ quæ-<lb/>dam curvæ EEE, FFF ità ad ſe relatæ, ut ductis rectis PEF ad <lb/>
<anchor type="note" xlink:label="note-0249-03a" xlink:href="note-0249-03"/>
poſitione datam BD parallelis, ſint ordinatæ PE ſemper ut quadrata <lb/>ex ordinatis PF; </s>
  <s xml:id="echoid-s11574" xml:space="preserve">rectæ verò ES, FT ( ex ejuſdem communis ordi-<lb/>natæ terminatis ductæ) curvas haſce contingant; </s>
  <s xml:id="echoid-s11575" xml:space="preserve">erit TP = 2 SP; <lb/></s>
  <s xml:id="echoid-s11576" xml:space="preserve">Quòd ſi ordinatæ PE ſe habeant ut ipſarum PF cubi, erit TP = 3 SP; </s>
  <s xml:id="echoid-s11577" xml:space="preserve"><lb/>ſi PE ſint ut quadrato quadrata ipſarum PF, erit TP = 4 SP; </s>
  <s xml:id="echoid-s11578" xml:space="preserve">ac <lb/>ſic eodem ad infinitum continuo tenore.</s>
  <s xml:id="echoid-s11579" xml:space="preserve"/>
</p>
<div xml:id="echoid-div337" type="float" level="2" n="7">
<note position="right" xlink:label="note-0249-03" xlink:href="note-0249-03a" xml:space="preserve">Fig. 96.</note>
</div>
<p>
  <s xml:id="echoid-s11580" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s11581" xml:space="preserve">Sit porrò Circulus ABC, cujus Centrum D, radius DB, <lb/>item lineæ EBE, FBF per B tranſeuntes, ac ità relatæ, ut ductâ <lb/>
<anchor type="note" xlink:label="note-0249-04a" xlink:href="note-0249-04"/>
per D rectâ quâpiam DG, ſit ſemper DF eodem ordine media Arith-<lb/>meticè inter DG, DE; </s>
  <s xml:id="echoid-s11582" xml:space="preserve">tangat autem recta BO curvam EBE in B; <lb/></s>
  <s xml:id="echoid-s11583" xml:space="preserve">oportet curvæ FBF tangentem (ad B) deſignare.</s>
  <s xml:id="echoid-s11584" xml:space="preserve"/>
</p>
<div xml:id="echoid-div338" type="float" level="2" n="8">
<note position="right" xlink:label="note-0249-04" xlink:href="note-0249-04a" xml:space="preserve">Fig. 97.</note>
</div>
<p>
  <s xml:id="echoid-s11585" xml:space="preserve">Hoc (certè <anchor type="note" xlink:href="" symbol="(_a_)"/> generatim quadantenus præſtitum) è re fuerit hîc
<anchor type="note" xlink:label="note-0249-05a" xlink:href="note-0249-05"/>
ſpeciatim apertiùs atque plenius exequi: </s>
  <s xml:id="echoid-s11586" xml:space="preserve">Quorſum ſit DQ ad DB <lb/>perpendicularis, quam ſecet BO in S; </s>
  <s xml:id="echoid-s11587" xml:space="preserve">fiat verò N. </s>
  <s xml:id="echoid-s11588" xml:space="preserve">M :</s>
  <s xml:id="echoid-s11589" xml:space="preserve">: DS. </s>
  <s xml:id="echoid-s11590" xml:space="preserve">DT; <lb/></s>
  <s xml:id="echoid-s11591" xml:space="preserve">connectatúrque recta TB; </s>
  <s xml:id="echoid-s11592" xml:space="preserve">hæc curvam FBF tanget.</s>
  <s xml:id="echoid-s11593" xml:space="preserve"/>
</p>
<div xml:id="echoid-div339" type="float" level="2" n="9">
<note position="right" xlink:label="note-0249-05" xlink:href="note-0249-05a" xml:space="preserve">a 8. Lect. 8.</note>
</div>
<p>
  <s xml:id="echoid-s11594" xml:space="preserve">Tangat enim recta PB _circulum_ AB G; </s>
  <s xml:id="echoid-s11595" xml:space="preserve">ſecentúrque rectæ D S <lb/>
<anchor type="note" xlink:label="note-0249-06a" xlink:href="note-0249-06"/>
in X, &amp; </s>
  <s xml:id="echoid-s11596" xml:space="preserve">BS in Y, ità ut ſit DS. </s>
  <s xml:id="echoid-s11597" xml:space="preserve">D X :</s>
  <s xml:id="echoid-s11598" xml:space="preserve">: M. </s>
  <s xml:id="echoid-s11599" xml:space="preserve">N :</s>
  <s xml:id="echoid-s11600" xml:space="preserve">: BS. </s>
  <s xml:id="echoid-s11601" xml:space="preserve">BY; </s>
  <s xml:id="echoid-s11602" xml:space="preserve">perque <lb/>puncta X, Y ducantur XZ ad BS, &amp; </s>
  <s xml:id="echoid-s11603" xml:space="preserve">YV ad DS parallelæ, concur-<lb/>rentes in C; </s>
  <s xml:id="echoid-s11604" xml:space="preserve">tum _aſymptotis_ YCZ per B traducta concipiatur _hyper_-<lb/>_bola_ LB L; </s>
  <s xml:id="echoid-s11605" xml:space="preserve">porrò ex D projiciatur utcunque recta DP dictas lineas
<pb o="72" file="0250" n="265" rhead=""/>
interſecans, ut expreſſum vides; </s>
  <s xml:id="echoid-s11606" xml:space="preserve">eſtque jam PK. </s>
  <s xml:id="echoid-s11607" xml:space="preserve">PL :</s>
  <s xml:id="echoid-s11608" xml:space="preserve">: <anchor type="note" xlink:href="" symbol="(_a_)"/> M. </s>
  <s xml:id="echoid-s11609" xml:space="preserve">N
<anchor type="note" xlink:label="note-0250-01a" xlink:href="note-0250-01"/>
:</s>
  <s xml:id="echoid-s11610" xml:space="preserve">: <anchor type="note" xlink:href="" symbol="(_b_)"/> GE. </s>
  <s xml:id="echoid-s11611" xml:space="preserve">GF <anchor type="note" xlink:href="" symbol="(_c_)"/> &amp;</s>
  <s xml:id="echoid-s11612" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11613" xml:space="preserve">PE. </s>
  <s xml:id="echoid-s11614" xml:space="preserve">PF &amp;</s>
  <s xml:id="echoid-s11615" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11616" xml:space="preserve">PK. </s>
  <s xml:id="echoid-s11617" xml:space="preserve">PF; </s>
  <s xml:id="echoid-s11618" xml:space="preserve">quare PL &amp;</s>
  <s xml:id="echoid-s11619" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11620" xml:space="preserve">PF;</s>
  <s xml:id="echoid-s11621" xml:space="preserve">
igitur _Hyperbola_ LBL curvam FB F tangit. </s>
  <s xml:id="echoid-s11622" xml:space="preserve">Protracta jam TB <lb/>
<anchor type="note" xlink:label="note-0250-02a" xlink:href="note-0250-02"/>
cum XZ conveniat in R; </s>
  <s xml:id="echoid-s11623" xml:space="preserve">eſtque tum RZ. </s>
  <s xml:id="echoid-s11624" xml:space="preserve">ZB :</s>
  <s xml:id="echoid-s11625" xml:space="preserve">: BS. </s>
  <s xml:id="echoid-s11626" xml:space="preserve">ST. </s>
  <s xml:id="echoid-s11627" xml:space="preserve">unde <lb/>
<anchor type="note" xlink:label="note-0250-03a" xlink:href="note-0250-03"/>
RZ xST = BS x ZB = BS x S X. </s>
  <s xml:id="echoid-s11628" xml:space="preserve">atqui propter DS. </s>
  <s xml:id="echoid-s11629" xml:space="preserve">SX :</s>
  <s xml:id="echoid-s11630" xml:space="preserve">: <anchor type="note" xlink:href="" symbol="(_d_)"/>
BS. </s>
  <s xml:id="echoid-s11631" xml:space="preserve">SY, eſt DS xSY = BS xSX. </s>
  <s xml:id="echoid-s11632" xml:space="preserve">ergò RZ xST = DS xS Y <lb/>
<anchor type="note" xlink:label="note-0250-04a" xlink:href="note-0250-04"/>
 = DS x CX. </s>
  <s xml:id="echoid-s11633" xml:space="preserve">vel RZ. </s>
  <s xml:id="echoid-s11634" xml:space="preserve">CX :</s>
  <s xml:id="echoid-s11635" xml:space="preserve">: DS . </s>
  <s xml:id="echoid-s11636" xml:space="preserve">ST; </s>
  <s xml:id="echoid-s11637" xml:space="preserve">compoſitéque RZ . </s>
  <s xml:id="echoid-s11638" xml:space="preserve">RZ <lb/>+ CX :</s>
  <s xml:id="echoid-s11639" xml:space="preserve">: DS. </s>
  <s xml:id="echoid-s11640" xml:space="preserve">DT :</s>
  <s xml:id="echoid-s11641" xml:space="preserve">: <anchor type="note" xlink:href="" symbol="(_d_)"/> N. </s>
  <s xml:id="echoid-s11642" xml:space="preserve">M :</s>
  <s xml:id="echoid-s11643" xml:space="preserve">: CZ. </s>
  <s xml:id="echoid-s11644" xml:space="preserve">CZ + CX. </s>
  <s xml:id="echoid-s11645" xml:space="preserve">itaque
diviſim eſt RZ. </s>
  <s xml:id="echoid-s11646" xml:space="preserve">CX :</s>
  <s xml:id="echoid-s11647" xml:space="preserve">: CZ. </s>
  <s xml:id="echoid-s11648" xml:space="preserve">CX. </s>
  <s xml:id="echoid-s11649" xml:space="preserve">adeóque RZ = CZ; </s>
  <s xml:id="echoid-s11650" xml:space="preserve">unde RB <lb/>_hyperbolam_ LBL tangit; </s>
  <s xml:id="echoid-s11651" xml:space="preserve">hæc igitur ( RBT) curvam FBF, ipſi <lb/>LBL contiguam, quoque tanget. </s>
  <s xml:id="echoid-s11652" xml:space="preserve">quod erat Propoſitum.</s>
  <s xml:id="echoid-s11653" xml:space="preserve"/>
</p>
<div xml:id="echoid-div340" type="float" level="2" n="10">
<note position="right" xlink:label="note-0249-06" xlink:href="note-0249-06a" xml:space="preserve">Fig. 97.</note>
<note position="left" xlink:label="note-0250-01" xlink:href="note-0250-01a" xml:space="preserve">_a Converſ_. 4. <lb/>Lect. VI.</note>
<note position="left" xlink:label="note-0250-02" xlink:href="note-0250-02a" xml:space="preserve">_b_ 11. Lect. VII.</note>
<note position="left" xlink:label="note-0250-03" xlink:href="note-0250-03a" xml:space="preserve">_c_ @@. Lect. VII.</note>
<note position="left" xlink:label="note-0250-04" xlink:href="note-0250-04a" xml:space="preserve">_d Conſtr_.</note>
</div>
<p>
  <s xml:id="echoid-s11654" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s11655" xml:space="preserve">Hinc ſi perſiſtentibus reliquis, recta tantùm DF jam inter <lb/>D G, DE perpetuò Geometricè media ſtatuatur ( eodem qui priùs fuit <lb/>ordine) eadem BT curvam FBF quoque continget.</s>
  <s xml:id="echoid-s11656" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11657" xml:space="preserve">Etenim ex mediis ejuſdem ordinis _Aritbmetice Geometricéque_ pro-<lb/>portionalibus efformatæ lineæ ſe mutuò contingunt, adeóque commu-<lb/>ni rectâ tangente gaudent.</s>
  <s xml:id="echoid-s11658" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11659" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s11660" xml:space="preserve">Porrò _(_ſtantibus reliquis ut in poſtremâ) quodvis in curva <lb/>
<anchor type="note" xlink:label="note-0250-05a" xlink:href="note-0250-05"/>
FB F deſignetur punctum F, quæ curvam ad hoc tanget recta ſimili <lb/>pacto determinatur.</s>
  <s xml:id="echoid-s11661" xml:space="preserve"/>
</p>
<div xml:id="echoid-div341" type="float" level="2" n="11">
<note position="left" xlink:label="note-0250-05" xlink:href="note-0250-05a" xml:space="preserve">Fig. 98.</note>
</div>
<p>
  <s xml:id="echoid-s11662" xml:space="preserve">Connectatur utique recta DF curvam EB E ſecans ad E ; </s>
  <s xml:id="echoid-s11663" xml:space="preserve">item du-<lb/>catur DQ ad DG perpendicularis ipſam EO interſecans ad X; </s>
  <s xml:id="echoid-s11664" xml:space="preserve">fiat <lb/>etiam DX. </s>
  <s xml:id="echoid-s11665" xml:space="preserve">DY :</s>
  <s xml:id="echoid-s11666" xml:space="preserve">: N. </s>
  <s xml:id="echoid-s11667" xml:space="preserve">M ; </s>
  <s xml:id="echoid-s11668" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11669" xml:space="preserve">connectatur EY; </s>
  <s xml:id="echoid-s11670" xml:space="preserve">ipſi demum EY pa-<lb/>rallela ducatur FZ; </s>
  <s xml:id="echoid-s11671" xml:space="preserve">hæc curvam FBF continget.</s>
  <s xml:id="echoid-s11672" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11673" xml:space="preserve">Nam centro D per E ducatur circulus CEI; </s>
  <s xml:id="echoid-s11674" xml:space="preserve">concipiatúrque linea <lb/>HEH talis, ut à D eductâ quacunque rectâ DI ( quæ circulum CE <lb/>ſecet in I, curvam HEH in H, &amp; </s>
  <s xml:id="echoid-s11675" xml:space="preserve">ipſam EB E in L ) ſit pepertuò <lb/>DH eodem inter DI, DL ordine proportionalis, quo DF inter DG, <lb/>DE; </s>
  <s xml:id="echoid-s11676" xml:space="preserve">palam eſt tunc (è præcedente) quòd recta EY curvam HEH <lb/>tanget; </s>
  <s xml:id="echoid-s11677" xml:space="preserve">verùm ipſi HEH <anchor type="note" xlink:href="" symbol="(_a_)"/> analoga eſt curva FBF; </s>
  <s xml:id="echoid-s11678" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_b_)"/> quare
<anchor type="note" xlink:label="note-0250-06a" xlink:href="note-0250-06"/>
recta FZ curvam FBF quoque tanget.</s>
  <s xml:id="echoid-s11679" xml:space="preserve"/>
</p>
<div xml:id="echoid-div342" type="float" level="2" n="12">
<note position="left" xlink:label="note-0250-06" xlink:href="note-0250-06a" xml:space="preserve">_a_ 9. Lect. VII.</note>
</div>
<note position="left" xml:space="preserve">_b_ 7. Lect. VIII.</note>
<p>
  <s xml:id="echoid-s11680" xml:space="preserve">Exhinc nedum innumerarum ſpiralium; </s>
  <s xml:id="echoid-s11681" xml:space="preserve">at aliarum diverſi generis <lb/>infinities plurium tangentes quàm promptè determinantur.</s>
  <s xml:id="echoid-s11682" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11683" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s11684" xml:space="preserve">Hinc clarum eſt, ſi duæ lineæ EEE, FEF ſic ad ſe referan-<lb/>tur, ut à puncto quodam D utcunque projectis rectis DEF; </s>
  <s xml:id="echoid-s11685" xml:space="preserve">habe-<lb/>
<anchor type="note" xlink:label="note-0250-08a" xlink:href="note-0250-08"/>
ant ſe rectæ DE, ut quadrata ex ipſis DF, &amp; </s>
  <s xml:id="echoid-s11686" xml:space="preserve">ad harum terminos <lb/>tangant curvas rectæ ES, FT; </s>
  <s xml:id="echoid-s11687" xml:space="preserve">cum perpendicularibus ad ipſas
<pb o="73" file="0251" n="266" rhead=""/>
DEF concurrentes punctis S, T; </s>
  <s xml:id="echoid-s11688" xml:space="preserve">erit ſemper DT = 2 DS. </s>
  <s xml:id="echoid-s11689" xml:space="preserve">Quòd <lb/>
<anchor type="note" xlink:label="note-0251-01a" xlink:href="note-0251-01"/>
ſi DE ſunt ut cubi ipſarum DF, erit ſemper DT = 3 DS; </s>
  <s xml:id="echoid-s11690" xml:space="preserve">ac ſi-<lb/>mili deinceps modo.</s>
  <s xml:id="echoid-s11691" xml:space="preserve"/>
</p>
<div xml:id="echoid-div343" type="float" level="2" n="13">
<note position="left" xlink:label="note-0250-08" xlink:href="note-0250-08a" xml:space="preserve">Fig. 99.</note>
<note position="right" xlink:label="note-0251-01" xlink:href="note-0251-01a" xml:space="preserve">Fig. 99.</note>
</div>
<p>
  <s xml:id="echoid-s11692" xml:space="preserve">X. </s>
  <s xml:id="echoid-s11693" xml:space="preserve">Sint rectæ VD, TB concurrentes in T, quas decuſſet poſnio-<lb/>
<anchor type="note" xlink:label="note-0251-02a" xlink:href="note-0251-02"/>
ne data recta DB; </s>
  <s xml:id="echoid-s11694" xml:space="preserve">tranſeant etiam per B lineæ EBE, FBF tales, <lb/>ut ductâ quâcunque PG ad DB parallelâ, ſit perpetuò PF eodem or-<lb/>dine media Arithmeticè inter PG, PE; </s>
  <s xml:id="echoid-s11695" xml:space="preserve">tangat autem BR curvam <lb/>EBE, opertet lineæ FBF tangentem ad B determinare.</s>
  <s xml:id="echoid-s11696" xml:space="preserve"/>
</p>
<div xml:id="echoid-div344" type="float" level="2" n="14">
<note position="right" xlink:label="note-0251-02" xlink:href="note-0251-02a" xml:space="preserve">Fig. 100</note>
</div>
<p>
  <s xml:id="echoid-s11697" xml:space="preserve">Sumptis NM (ordinum in quibus ſunt PF, PE exponentibus) <lb/>fiat N x TD + M \\ - N} x RD. </s>
  <s xml:id="echoid-s11698" xml:space="preserve">M x TD:</s>
  <s xml:id="echoid-s11699" xml:space="preserve">: RD. </s>
  <s xml:id="echoid-s11700" xml:space="preserve">SD; </s>
  <s xml:id="echoid-s11701" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11702" xml:space="preserve">connecta-<lb/>tur BS; </s>
  <s xml:id="echoid-s11703" xml:space="preserve">hæc curvam FBF continget.</s>
  <s xml:id="echoid-s11704" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11705" xml:space="preserve">Nam utcunque ducta ſit PG, dictas lineas ſecans ut vides. </s>
  <s xml:id="echoid-s11706" xml:space="preserve">Eſtque <lb/>EG. </s>
  <s xml:id="echoid-s11707" xml:space="preserve">FG:</s>
  <s xml:id="echoid-s11708" xml:space="preserve">: <anchor type="note" xlink:href="" symbol="(_a_)"/> M. </s>
  <s xml:id="echoid-s11709" xml:space="preserve">N. </s>
  <s xml:id="echoid-s11710" xml:space="preserve">ergò FG x TD. </s>
  <s xml:id="echoid-s11711" xml:space="preserve">EG x TD:</s>
  <s xml:id="echoid-s11712" xml:space="preserve">: N x TD.</s>
  <s xml:id="echoid-s11713" xml:space="preserve">
M x TD. </s>
  <s xml:id="echoid-s11714" xml:space="preserve">Item EF x RD. </s>
  <s xml:id="echoid-s11715" xml:space="preserve">EG x TD:</s>
  <s xml:id="echoid-s11716" xml:space="preserve">: M - N x RD. </s>
  <s xml:id="echoid-s11717" xml:space="preserve">M x <lb/>
<anchor type="note" xlink:label="note-0251-03a" xlink:href="note-0251-03"/>
TD. </s>
  <s xml:id="echoid-s11718" xml:space="preserve">Quapropter (antecedentes conjungendo) erit FG x TD + <lb/>EF x RD. </s>
  <s xml:id="echoid-s11719" xml:space="preserve">EG x TD:</s>
  <s xml:id="echoid-s11720" xml:space="preserve">: N x TD + M - N x RD. </s>
  <s xml:id="echoid-s11721" xml:space="preserve">M x TD; <lb/></s>
  <s xml:id="echoid-s11722" xml:space="preserve">(hoc eſt):</s>
  <s xml:id="echoid-s11723" xml:space="preserve">: <anchor type="note" xlink:href="" symbol="(_b_)"/> RD. </s>
  <s xml:id="echoid-s11724" xml:space="preserve">SD. </s>
  <s xml:id="echoid-s11725" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_c_)"/> Eſt antem LG x TD + KL x RD.</s>
  <s xml:id="echoid-s11726" xml:space="preserve">
<anchor type="note" xlink:label="note-0251-04a" xlink:href="note-0251-04"/>
<anchor type="note" xlink:label="note-0251-05a" xlink:href="note-0251-05"/>
KG x TD:</s>
  <s xml:id="echoid-s11727" xml:space="preserve">: RD. </s>
  <s xml:id="echoid-s11728" xml:space="preserve">SD. </s>
  <s xml:id="echoid-s11729" xml:space="preserve">quare FG x TD + EF x RD. </s>
  <s xml:id="echoid-s11730" xml:space="preserve">EG x <lb/>TD:</s>
  <s xml:id="echoid-s11731" xml:space="preserve">: LG x TD + KL x RD. </s>
  <s xml:id="echoid-s11732" xml:space="preserve">KG x TD. </s>
  <s xml:id="echoid-s11733" xml:space="preserve">hinc, cùm ſit EG <anchor type="note" xlink:href="" symbol="(_d_)"/>
<anchor type="note" xlink:label="note-0251-06a" xlink:href="note-0251-06"/>
&amp;</s>
  <s xml:id="echoid-s11734" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11735" xml:space="preserve">KG; </s>
  <s xml:id="echoid-s11736" xml:space="preserve">erit FG x TD + EF x RD &amp;</s>
  <s xml:id="echoid-s11737" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11738" xml:space="preserve">LG x TD + KL x RD; <lb/></s>
  <s xml:id="echoid-s11739" xml:space="preserve">vel FG. </s>
  <s xml:id="echoid-s11740" xml:space="preserve">EF + TD. </s>
  <s xml:id="echoid-s11741" xml:space="preserve">RD &amp;</s>
  <s xml:id="echoid-s11742" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11743" xml:space="preserve">LG. </s>
  <s xml:id="echoid-s11744" xml:space="preserve">KL + TD. </s>
  <s xml:id="echoid-s11745" xml:space="preserve">RD; </s>
  <s xml:id="echoid-s11746" xml:space="preserve">ſeu (dem-<lb/>ptâ communi ratione) FG. </s>
  <s xml:id="echoid-s11747" xml:space="preserve">EF &amp;</s>
  <s xml:id="echoid-s11748" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11749" xml:space="preserve">LG. </s>
  <s xml:id="echoid-s11750" xml:space="preserve">KL. </s>
  <s xml:id="echoid-s11751" xml:space="preserve">vel componendo <lb/>EG. </s>
  <s xml:id="echoid-s11752" xml:space="preserve">EF &amp;</s>
  <s xml:id="echoid-s11753" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11754" xml:space="preserve">KG. </s>
  <s xml:id="echoid-s11755" xml:space="preserve">KL <anchor type="note" xlink:href="" symbol="(_e_)"/> &amp;</s>
  <s xml:id="echoid-s11756" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11757" xml:space="preserve">EG. </s>
  <s xml:id="echoid-s11758" xml:space="preserve">EL. </s>
  <s xml:id="echoid-s11759" xml:space="preserve">unde eſt EF &amp;</s>
  <s xml:id="echoid-s11760" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11761" xml:space="preserve">EL.</s>
  <s xml:id="echoid-s11762" xml:space="preserve">
<anchor type="note" xlink:label="note-0251-07a" xlink:href="note-0251-07"/>
itaque punctum L extra curvam FBF ſitum eſt; </s>
  <s xml:id="echoid-s11763" xml:space="preserve">adeoque liquet <lb/>Propoſitum.</s>
  <s xml:id="echoid-s11764" xml:space="preserve"/>
</p>
<div xml:id="echoid-div345" type="float" level="2" n="15">
<note symbol="(_a_)" position="right" xlink:label="note-0251-03" xlink:href="note-0251-03a" xml:space="preserve">11. Lect. <lb/>VII.</note>
<note symbol="(_b_)" position="right" xlink:label="note-0251-04" xlink:href="note-0251-04a" xml:space="preserve">_Conſtr._</note>
<note symbol="(_c_)" position="right" xlink:label="note-0251-05" xlink:href="note-0251-05a" xml:space="preserve">4. Lect. <lb/>VII.</note>
<note symbol="(_d_)" position="right" xlink:label="note-0251-06" xlink:href="note-0251-06a" xml:space="preserve">_Hyp_</note>
<note symbol="(_e_)" position="right" xlink:label="note-0251-07" xlink:href="note-0251-07a" xml:space="preserve">1. Lect. <lb/>VII.</note>
</div>
<p>
  <s xml:id="echoid-s11765" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s11766" xml:space="preserve">Quinetiam, reliquis ſtantibus iiſdem, ſi PF ſupponatur ejuſ-<lb/>dem ordinis Geometricè media liquet (planè ſicut in modò præceden-<lb/>tibus) eandem BS curvam FBF contingere.</s>
  <s xml:id="echoid-s11767" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11768" xml:space="preserve">_Exemplnum._ </s>
  <s xml:id="echoid-s11769" xml:space="preserve">Si PF ſit è ſex mediis tertia, ſeu M = 7; </s>
  <s xml:id="echoid-s11770" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11771" xml:space="preserve">N = 3; <lb/></s>
  <s xml:id="echoid-s11772" xml:space="preserve">erit 3 TD + 4 RD. </s>
  <s xml:id="echoid-s11773" xml:space="preserve">7 MD:</s>
  <s xml:id="echoid-s11774" xml:space="preserve">: RD. </s>
  <s xml:id="echoid-s11775" xml:space="preserve">SD; </s>
  <s xml:id="echoid-s11776" xml:space="preserve">vel SD = {7 MD x RD/3 TD + 4 RD.</s>
  <s xml:id="echoid-s11777" xml:space="preserve">}</s>
</p>
<p>
  <s xml:id="echoid-s11778" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s11779" xml:space="preserve">Patet etiam, accepto quolibet in curva FBF puncto (ceu F) <lb/>rectam ad hoc tangentem conſimili pacto deſignari. </s>
  <s xml:id="echoid-s11780" xml:space="preserve">Nempe per F <lb/>
<anchor type="note" xlink:label="note-0251-08a" xlink:href="note-0251-08"/>
ducatur recta PG ad DB parallela, ſecans curvam EBE ad E; </s>
  <s xml:id="echoid-s11781" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11782" xml:space="preserve"><lb/>per E ducatur ER curvam EBE tangens; </s>
  <s xml:id="echoid-s11783" xml:space="preserve">fiátque N x TP + M/- N} x RP.</s>
  <s xml:id="echoid-s11784" xml:space="preserve">
<pb o="74" file="0252" n="267" rhead=""/>
M x TP:</s>
  <s xml:id="echoid-s11785" xml:space="preserve">: RP. </s>
  <s xml:id="echoid-s11786" xml:space="preserve">SP; </s>
  <s xml:id="echoid-s11787" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11788" xml:space="preserve">connectatur SF; </s>
  <s xml:id="echoid-s11789" xml:space="preserve">hæc curvam <lb/>FBF tanget; </s>
  <s xml:id="echoid-s11790" xml:space="preserve">id quod omnino ſimili diſcurſu demonſtratur, quo ter-<lb/>tia hujus; </s>
  <s xml:id="echoid-s11791" xml:space="preserve">tantùm hîc (non per E ad VD parallela ducitur, at) con-<lb/>nectitur ET; </s>
  <s xml:id="echoid-s11792" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11793" xml:space="preserve">loco ſeptimæ allegatur octava ſeptimæ Lectionis. <lb/></s>
  <s xml:id="echoid-s11794" xml:space="preserve">quid plura?</s>
  <s xml:id="echoid-s11795" xml:space="preserve"/>
</p>
<div xml:id="echoid-div346" type="float" level="2" n="16">
<note position="right" xlink:label="note-0251-08" xlink:href="note-0251-08a" xml:space="preserve">Fig. 101.</note>
</div>
<p>
  <s xml:id="echoid-s11796" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s11797" xml:space="preserve">Adnotetur, ſi linea EBE ſit recta, (rectæ nempe BR coin-<lb/>cidens) eſſe lineam FBF ex _infinitis hyperbolis_ (vel _hyperboliformi-_ <lb/>_bus_) aliquam; </s>
  <s xml:id="echoid-s11798" xml:space="preserve">quarum igitur (unà cùm aliarum infinities diverſi ge-<lb/>neris plurium) _Tangentes_ determinandi modum uno _Tbeorem<unsure/>ate_ com-<lb/>plexi ſumus.</s>
  <s xml:id="echoid-s11799" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11800" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s11801" xml:space="preserve">Quòd ſi puncta T, R non ad eaſdem partes puncti D (vel P) <lb/>cadant; </s>
  <s xml:id="echoid-s11802" xml:space="preserve">curvæ FBF tangens (BS) deſignatur faciendo N x RD-: <lb/></s>
  <s xml:id="echoid-s11803" xml:space="preserve">
<anchor type="note" xlink:label="note-0252-01a" xlink:href="note-0252-01"/>
M \\ - N} x TD. </s>
  <s xml:id="echoid-s11804" xml:space="preserve">M x TD:</s>
  <s xml:id="echoid-s11805" xml:space="preserve">: RD. </s>
  <s xml:id="echoid-s11806" xml:space="preserve">SD.</s>
  <s xml:id="echoid-s11807" xml:space="preserve"/>
</p>
<div xml:id="echoid-div347" type="float" level="2" n="17">
<note position="left" xlink:label="note-0252-01" xlink:href="note-0252-01a" xml:space="preserve">Fig. 102.</note>
</div>
<p>
  <s xml:id="echoid-s11808" xml:space="preserve">Simili planè diſcurſu conſtat hoc, tantùm (quartæ loco) ſeptimæ <lb/>Lectionis quintam adhibendo.</s>
  <s xml:id="echoid-s11809" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11810" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s11811" xml:space="preserve">Hinc autem nedum _Ellipſoidum_ omnium (poſito nempe line-<lb/>am EBE rectam eſſe, lineæ BR coincidentem) aſt aliarum alterius <lb/>generis _linearnm innumer abilium Taxgentes_ unâ operâ determinan-<lb/>tur.</s>
  <s xml:id="echoid-s11812" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11813" xml:space="preserve">_Exemplum._ </s>
  <s xml:id="echoid-s11814" xml:space="preserve">Si PF ſit è quatuor mediis quarta, ſeu M = 5; </s>
  <s xml:id="echoid-s11815" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11816" xml:space="preserve">N <lb/> = 4; </s>
  <s xml:id="echoid-s11817" xml:space="preserve">erit SD = {5 TD x RD/4 RD - TD.</s>
  <s xml:id="echoid-s11818" xml:space="preserve">}</s>
</p>
<p>
  <s xml:id="echoid-s11819" xml:space="preserve">_Notetur_; </s>
  <s xml:id="echoid-s11820" xml:space="preserve">Si contigerit eſſe ND x RD = M/- N} x TD, eſſe DS <lb/>infinitam; </s>
  <s xml:id="echoid-s11821" xml:space="preserve">ſeu BS ipſi VD parallelam. </s>
  <s xml:id="echoid-s11822" xml:space="preserve">Alia poſſent adnotari; </s>
  <s xml:id="echoid-s11823" xml:space="preserve">ſed <lb/>relinquo.</s>
  <s xml:id="echoid-s11824" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11825" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s11826" xml:space="preserve">Inter alias curvas innumeras, etiam hâc methodo _Ciſſois_ &amp; </s>
  <s xml:id="echoid-s11827" xml:space="preserve"><lb/>_Ciſſoidaliam_ omne genus comprehenditur: </s>
  <s xml:id="echoid-s11828" xml:space="preserve">Sit utique ſemirectus an-<lb/>
<anchor type="note" xlink:label="note-0252-02a" xlink:href="note-0252-02"/>
gulus DSB; </s>
  <s xml:id="echoid-s11829" xml:space="preserve">curvæque duæ SGB, SEE ſic ad ſe referantur, ut <lb/>ductâ liberè rectâ GE ad BD parallelâ, (quæ lineas expoſitas, ut <lb/>conſpicis, ſecet) ſint PG, PF, PE continuè proportionales; </s>
  <s xml:id="echoid-s11830" xml:space="preserve">tangat <lb/>autem recta GT curvam SGB in G, reperietur quæ ad E lineam SEB <lb/>tangit, faciendo 2 TP - SP. </s>
  <s xml:id="echoid-s11831" xml:space="preserve">TP:</s>
  <s xml:id="echoid-s11832" xml:space="preserve">: SP. </s>
  <s xml:id="echoid-s11833" xml:space="preserve">RP; </s>
  <s xml:id="echoid-s11834" xml:space="preserve">utique connexa <lb/>RE curvam SEE tanget. </s>
  <s xml:id="echoid-s11835" xml:space="preserve">Id quod è præmiſſis facilè colligitur. <lb/></s>
  <s xml:id="echoid-s11836" xml:space="preserve">Quòd ſi jam curva SGB ſit circulus, &amp; </s>
  <s xml:id="echoid-s11837" xml:space="preserve">applicationis angulus SPG
<pb o="75" file="0253" n="268" rhead=""/>
ſit rectus, erit curva SEE _Ciſſois vulgaris_, ſeu _Dioslea_; </s>
  <s xml:id="echoid-s11838" xml:space="preserve">alioquin <lb/>alterius generis _Ciſſoidalis_. </s>
  <s xml:id="echoid-s11839" xml:space="preserve">Hoc autem ἐγ παςόδφ perſtringo. </s>
  <s xml:id="echoid-s11840" xml:space="preserve">Neq; <lb/></s>
  <s xml:id="echoid-s11841" xml:space="preserve">jam ampliùs vos detinebo.</s>
  <s xml:id="echoid-s11842" xml:space="preserve"/>
</p>
<div xml:id="echoid-div348" type="float" level="2" n="18">
<note position="left" xlink:label="note-0252-02" xlink:href="note-0252-02a" xml:space="preserve">Fig. 103.</note>
</div>
</div>
<div xml:id="echoid-div350" type="section" level="1" n="35">
<head xml:id="echoid-head38" xml:space="preserve"><emph style="sc">Lect</emph>. X.</head>
<p>
  <s xml:id="echoid-s11843" xml:space="preserve">IN ſtitutum circa tangentes negotium adhuc urgeo.</s>
  <s xml:id="echoid-s11844" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11845" xml:space="preserve">I. </s>
  <s xml:id="echoid-s11846" xml:space="preserve">Sit curva quæpiam AEG, nec non alia AFI ſic ad illam rela-<lb/>
<anchor type="note" xlink:label="note-0253-01a" xlink:href="note-0253-01"/>
ta, ut ductâ quâcunque EF ad poſitione datam AB parallelâ (quæ <lb/>curvam AFG ſecet in E, curvámque AFI in F (ſit perpetim EF <lb/>æqualis curvæ AEG ab A intercepto arcui AE; </s>
  <s xml:id="echoid-s11847" xml:space="preserve">tangat autem recta <lb/>ET curvam AEG in E, ſitque ET æqualis arcui AE, &amp; </s>
  <s xml:id="echoid-s11848" xml:space="preserve">connecta-<lb/>tur recta TF; </s>
  <s xml:id="echoid-s11849" xml:space="preserve">hæc curvam AFI tanget.</s>
  <s xml:id="echoid-s11850" xml:space="preserve"/>
</p>
<div xml:id="echoid-div350" type="float" level="2" n="1">
<note position="right" xlink:label="note-0253-01" xlink:href="note-0253-01a" xml:space="preserve">Fig. 104.</note>
</div>
<p>
  <s xml:id="echoid-s11851" xml:space="preserve">Nam ducatur ntcunque recta GK ad AB parallela, lineas propo-<lb/>ſitas ſecans, ut cernis; </s>
  <s xml:id="echoid-s11852" xml:space="preserve">éſtque GK = GH + HK = GH + HT <lb/>
<anchor type="note" xlink:href="" symbol="(_a_)"/> &amp;</s>
  <s xml:id="echoid-s11853" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11854" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s11855" xml:space="preserve">AG = GI; </s>
  <s xml:id="echoid-s11856" xml:space="preserve">unde punctum K extra curvam AFI ſi-
<anchor type="note" xlink:label="note-0253-02a" xlink:href="note-0253-02"/>
tum eſt; </s>
  <s xml:id="echoid-s11857" xml:space="preserve">adeóque recta TK ipſam tangit.</s>
  <s xml:id="echoid-s11858" xml:space="preserve"/>
</p>
<div xml:id="echoid-div351" type="float" level="2" n="2">
<note symbol="(_a_)" position="right" xlink:label="note-0253-02" xlink:href="note-0253-02a" xml:space="preserve">22 Lect. <lb/>VII.</note>
</div>
<p>
  <s xml:id="echoid-s11859" xml:space="preserve">II. </s>
  <s xml:id="echoid-s11860" xml:space="preserve">Quòd ſi recta EF quamlibet ad arcum AE rationem ſemper <lb/>eandem habeat, nihilo ſeciùs recta FT curvam AFI tanget; </s>
  <s xml:id="echoid-s11861" xml:space="preserve">ut ex <lb/>hac, &amp; </s>
  <s xml:id="echoid-s11862" xml:space="preserve">octavæ Lectionis ſexta manifeſtæ conſectatur.</s>
  <s xml:id="echoid-s11863" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11864" xml:space="preserve">Hæc antea pridem aliter oſtendimus; </s>
  <s xml:id="echoid-s11865" xml:space="preserve">aſt hæc demonſtratio ſimpli-<lb/>cior aliquanto videtur, &amp; </s>
  <s xml:id="echoid-s11866" xml:space="preserve">clarior; </s>
  <s xml:id="echoid-s11867" xml:space="preserve">methodóque quam inſinuamus ac-<lb/>commodatior.</s>
  <s xml:id="echoid-s11868" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11869" xml:space="preserve">III. </s>
  <s xml:id="echoid-s11870" xml:space="preserve">Sit _curva_ quæpiam AGE, punctúmque deſignatum D; </s>
  <s xml:id="echoid-s11871" xml:space="preserve">ſit <lb/>item alia curva AIF talis, ut à D projectâ rectâ quâ cunque DEF, <lb/>
<anchor type="note" xlink:label="note-0253-03a" xlink:href="note-0253-03"/>
ſit ſemper intercepta EF par arcui AE; </s>
  <s xml:id="echoid-s11872" xml:space="preserve">tangátque recta ET curvam <lb/>AGE; </s>
  <s xml:id="echoid-s11873" xml:space="preserve">oportet curvæ AIF _Tangentem_ (ad F) deſignare.</s>
  <s xml:id="echoid-s11874" xml:space="preserve"/>
</p>
<div xml:id="echoid-div352" type="float" level="2" n="3">
<note position="right" xlink:label="note-0253-03" xlink:href="note-0253-03a" xml:space="preserve">Fig. 105.</note>
</div>
<p>
  <s xml:id="echoid-s11875" xml:space="preserve">Fiat TE = arc. </s>
  <s xml:id="echoid-s11876" xml:space="preserve">AE; </s>
  <s xml:id="echoid-s11877" xml:space="preserve">ſitque curva TKF talis, ut ductâ utcunque <lb/>(è D) rectâ DK (quæ curvam TKF ſecet in K, rectámque TE in H)
<pb o="76" file="0254" n="269" rhead=""/>
ſit ſemper HK = HT; </s>
  <s xml:id="echoid-s11878" xml:space="preserve">tum curvam TKF <anchor type="note" xlink:href="" symbol="(_a_)"/> tangat recta FS in F;</s>
  <s xml:id="echoid-s11879" xml:space="preserve">
<anchor type="note" xlink:label="note-0254-01a" xlink:href="note-0254-01"/>
hæc curvam AIF quoque continget.</s>
  <s xml:id="echoid-s11880" xml:space="preserve"/>
</p>
<div xml:id="echoid-div353" type="float" level="2" n="4">
<note symbol="(_a_)" position="left" xlink:label="note-0254-01" xlink:href="note-0254-01a" xml:space="preserve">17. Lect. <lb/>VIII.</note>
</div>
<p>
  <s xml:id="echoid-s11881" xml:space="preserve">Eſt enim GK = GH + HK = GH + HT <anchor type="note" xlink:href="" symbol="(_a_)"/> GA = GI.</s>
  <s xml:id="echoid-s11882" xml:space="preserve">
<anchor type="note" xlink:label="note-0254-02a" xlink:href="note-0254-02"/>
quare punctum K extra curvam AIF jacet; </s>
  <s xml:id="echoid-s11883" xml:space="preserve">adeóque recta FS cur-<lb/>vam AIF continget.</s>
  <s xml:id="echoid-s11884" xml:space="preserve"/>
</p>
<div xml:id="echoid-div354" type="float" level="2" n="5">
<note symbol="(_a_)" position="left" xlink:label="note-0254-02" xlink:href="note-0254-02a" xml:space="preserve">22. Lect. <lb/>VII.</note>
</div>
<p>
  <s xml:id="echoid-s11885" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s11886" xml:space="preserve">Quòd ſi recta EF ad arcum AE eandem aliquamcunque ſtatu-<lb/>atur habere proportionem, tangens ejus facilè determinatur ex hac, &amp; </s>
  <s xml:id="echoid-s11887" xml:space="preserve"><lb/>octava octavæ Lectionis.</s>
  <s xml:id="echoid-s11888" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11889" xml:space="preserve">V. </s>
  <s xml:id="echoid-s11890" xml:space="preserve">Sint recta AP, duæque _curvæ_ AEG, AFI, ità ad ſe relatæ <lb/>
<anchor type="note" xlink:label="note-0254-03a" xlink:href="note-0254-03"/>
ut ductâ utcunque rectâ DEF (quæ rectam AP, curvas AEG, <lb/>AFI punctis D, E, F, ſecet) ſit ſemper recta DT æqualis arcui AE; <lb/></s>
  <s xml:id="echoid-s11891" xml:space="preserve">tangat autem recta ET curvam AEG ad E; </s>
  <s xml:id="echoid-s11892" xml:space="preserve">ſumatúrque ET par <lb/>arcui EA; </s>
  <s xml:id="echoid-s11893" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11894" xml:space="preserve">ſit TR ad BA parallela; </s>
  <s xml:id="echoid-s11895" xml:space="preserve">connectatur denuò recta RF; </s>
  <s xml:id="echoid-s11896" xml:space="preserve"><lb/>hæc curvam AFI tanget.</s>
  <s xml:id="echoid-s11897" xml:space="preserve"/>
</p>
<div xml:id="echoid-div355" type="float" level="2" n="6">
<note position="left" xlink:label="note-0254-03" xlink:href="note-0254-03a" xml:space="preserve">Fig. 106.</note>
</div>
<p>
  <s xml:id="echoid-s11898" xml:space="preserve">Concipiatur enim curva LFL talis; </s>
  <s xml:id="echoid-s11899" xml:space="preserve">ut ductâ quâcunque rectâ PL <lb/>
<anchor type="note" xlink:label="note-0254-04a" xlink:href="note-0254-04"/>
ad AB parallelâ (quæ curvam AEG in G, rectam TE in H, cur-<lb/>
<anchor type="note" xlink:label="note-0254-05a" xlink:href="note-0254-05"/>
vam LFL in L ſecet) ſit perpetuò recta PL æqualis ipſis TH, HG <lb/>ſimul; </s>
  <s xml:id="echoid-s11900" xml:space="preserve">eſt itaque PL <anchor type="note" xlink:href="" symbol="(_a_)"/> &amp;</s>
  <s xml:id="echoid-s11901" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11902" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s11903" xml:space="preserve">AEG * = PI. </s>
  <s xml:id="echoid-s11904" xml:space="preserve">Unde curva LFL
<anchor type="note" xlink:label="note-0254-06a" xlink:href="note-0254-06"/>
curvam AFI<unsure/> tangit. </s>
  <s xml:id="echoid-s11905" xml:space="preserve">Item recta IK <anchor type="note" xlink:href="" symbol="(_b_)"/> æquatur rectæ TH; </s>
  <s xml:id="echoid-s11906" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_c_)"/>
<anchor type="note" xlink:label="note-0254-07a" xlink:href="note-0254-07"/>
adeóque curva LFL rectam RFK tangit; </s>
  <s xml:id="echoid-s11907" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_d_)"/> quare curvam AFI
<anchor type="note" xlink:label="note-0254-08a" xlink:href="note-0254-08"/>
tanget recta.</s>
  <s xml:id="echoid-s11908" xml:space="preserve"/>
</p>
<div xml:id="echoid-div356" type="float" level="2" n="7">
<note symbol="(_a_)" position="left" xlink:label="note-0254-04" xlink:href="note-0254-04a" xml:space="preserve">22. Lect. <lb/>VII.</note>
<note symbol="(_b_)" position="left" xlink:label="note-0254-05" xlink:href="note-0254-05a" xml:space="preserve">26. Lect. <lb/>VI.</note>
<note symbol="* it" position="left" xlink:label="note-0254-06" xlink:href="note-0254-06a" xml:space="preserve">Hyp.</note>
<note symbol="(_c_)" position="left" xlink:label="note-0254-07" xlink:href="note-0254-07a" xml:space="preserve">3 Lect. <lb/>VIII.</note>
<note symbol="(_d_)" position="left" xlink:label="note-0254-08" xlink:href="note-0254-08a" xml:space="preserve">2. Lect. <lb/>VIII.</note>
</div>
<p>
  <s xml:id="echoid-s11909" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s11910" xml:space="preserve">Etiam ſi rectæ DE ad arcus AE quamlibet ſemper eandem ra-<lb/>tionem habeant, recta RF nihilominus curvam AFI tanget, ut <lb/>ex hac, &amp; </s>
  <s xml:id="echoid-s11911" xml:space="preserve">ſexta octavæ Lectionis facilè patet.</s>
  <s xml:id="echoid-s11912" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11913" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s11914" xml:space="preserve">Sit punctum D; </s>
  <s xml:id="echoid-s11915" xml:space="preserve">duæque curvæ AGE, DIF itâ verſus ſe <lb/>
<anchor type="note" xlink:label="note-0254-09a" xlink:href="note-0254-09"/>
relatæ ſint, ut à puncto D projectâ quâvis rectâ DFE, ſit perpetuò <lb/>recta DF æqualis arcui AE; </s>
  <s xml:id="echoid-s11916" xml:space="preserve">tangat autem recta ET curvam AGE <lb/>ad E; </s>
  <s xml:id="echoid-s11917" xml:space="preserve">deſignanda jam eſt recta, quæ curvam DIF tangat (ad F).</s>
  <s xml:id="echoid-s11918" xml:space="preserve"/>
</p>
<div xml:id="echoid-div357" type="float" level="2" n="8">
<note position="left" xlink:label="note-0254-09" xlink:href="note-0254-09a" xml:space="preserve">Fig. 107.</note>
</div>
<p>
  <s xml:id="echoid-s11919" xml:space="preserve">Sumatur ET par _arcui_ FS; </s>
  <s xml:id="echoid-s11920" xml:space="preserve">concipiatúrque _carva_ DKK talis, ut <lb/>à D projectâ utcunque rectâ DH (quæ curvam DKK in K, rectam <lb/>
<anchor type="note" xlink:label="note-0254-10a" xlink:href="note-0254-10"/>
TE in H ſecet) ſit perpetuò DK = TH; </s>
  <s xml:id="echoid-s11921" xml:space="preserve">tum curvam DKK <anchor type="note" xlink:href="" symbol="(_a_)"/>
tangat recta FS ad F; </s>
  <s xml:id="echoid-s11922" xml:space="preserve">hæc curvam DIF quoque tanget.</s>
  <s xml:id="echoid-s11923" xml:space="preserve"/>
</p>
<div xml:id="echoid-div358" type="float" level="2" n="9">
<note symbol="(_a_)" position="left" xlink:label="note-0254-10" xlink:href="note-0254-10a" xml:space="preserve">16. Lect. <lb/>VIII.</note>
</div>
<p>
  <s xml:id="echoid-s11924" xml:space="preserve">Intelligatur enim _curva_ LFL talis, ut à D projectâ quapiam rectâ <lb/>
<anchor type="note" xlink:label="note-0254-11a" xlink:href="note-0254-11"/>
DH (quæ rectam TE ſecet in H, curvam LFL in L) ſit ſemper <lb/>
<anchor type="note" xlink:label="note-0254-12a" xlink:href="note-0254-12"/>
DL = TH + HG; </s>
  <s xml:id="echoid-s11925" xml:space="preserve">eſt itaque DL <anchor type="note" xlink:href="" symbol="(_b_)"/> &amp;</s>
  <s xml:id="echoid-s11926" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11927" xml:space="preserve">are. </s>
  <s xml:id="echoid-s11928" xml:space="preserve">AG <anchor type="note" xlink:href="" symbol="(_c_)"/> = DI,
<anchor type="note" xlink:label="note-0254-13a" xlink:href="note-0254-13"/>
<anchor type="note" xlink:href="" symbol="(_d_)"/> itaque curvæ DIF, LFL ſeſe <anchor type="note" xlink:href="" symbol="(_b_)"/> contingent, item curvæ KFK,
<pb o="77" file="0255" n="270" rhead=""/>
LFK ſeſe contingunt. </s>
  <s xml:id="echoid-s11929" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_e_)"/> quare curvæ DIF, KFK ſe quoque con-
<anchor type="note" xlink:label="note-0255-01a" xlink:href="note-0255-01"/>
tingent. </s>
  <s xml:id="echoid-s11930" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_e_)"/> ergò denique recta FS curvam DIF continget.</s>
  <s xml:id="echoid-s11931" xml:space="preserve"/>
</p>
<div xml:id="echoid-div359" type="float" level="2" n="10">
<note symbol="(_b_)" position="left" xlink:label="note-0254-11" xlink:href="note-0254-11a" xml:space="preserve">22. Lect. <lb/>VII.</note>
<note symbol="(_c_) it" position="left" xlink:label="note-0254-12" xlink:href="note-0254-12a" xml:space="preserve">Hyp.</note>
<note symbol="(_d_)" position="left" xlink:label="note-0254-13" xlink:href="note-0254-13a" xml:space="preserve">4. Lect. <lb/>VIII.</note>
<note symbol="(_e_)" position="right" xlink:label="note-0255-01" xlink:href="note-0255-01a" xml:space="preserve">2. Lect. <lb/>VIII.</note>
</div>
<p>
  <s xml:id="echoid-s11932" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s11933" xml:space="preserve">Quòd ſi rectæ DF quamvis aliam conſtanter eandem ad ar-<lb/>cus AE rationem obtinuerint, itidem deſignari poteſt recta curvam <lb/>DIF tangens, ex hac, &amp; </s>
  <s xml:id="echoid-s11934" xml:space="preserve">ſeptima octavæ Lectionis; </s>
  <s xml:id="echoid-s11935" xml:space="preserve">erit utique tan-<lb/>gens iſta huic FS parallela.</s>
  <s xml:id="echoid-s11936" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11937" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s11938" xml:space="preserve">Hinc nedum _ſpiralis circularis_, aſt innumerabilium ſimili ratione <lb/>progenitarum aliarum curvarum _Tangentes_ determinantur.</s>
  <s xml:id="echoid-s11939" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11940" xml:space="preserve">X. </s>
  <s xml:id="echoid-s11941" xml:space="preserve">Sint curva quæpiam AEH, recta AD (in qua determinatum <lb/>
<anchor type="note" xlink:label="note-0255-02a" xlink:href="note-0255-02"/>
punctum D) recta DH poſitione data; </s>
  <s xml:id="echoid-s11942" xml:space="preserve">ſit item curva AGB talis, <lb/>ut in hac aſſumpto quocunque puncto G, &amp; </s>
  <s xml:id="echoid-s11943" xml:space="preserve">per hoc ac D projectâ <lb/>rectâ DGE (quæ curvam AEH ſecet in E) ductâque GF ad DH <lb/>parallelâ habeant AE, AF aſſignatam rationem X ad Y; </s>
  <s xml:id="echoid-s11944" xml:space="preserve">tangat au-<lb/>tem recta ET curvam AEH; </s>
  <s xml:id="echoid-s11945" xml:space="preserve">recta deſignetur oportet, quæ curvam <lb/>AGB ad G tangat.</s>
  <s xml:id="echoid-s11946" xml:space="preserve"/>
</p>
<div xml:id="echoid-div360" type="float" level="2" n="11">
<note position="right" xlink:label="note-0255-02" xlink:href="note-0255-02a" xml:space="preserve">Fig. 108</note>
</div>
<p>
  <s xml:id="echoid-s11947" xml:space="preserve">Fiat recta EV æqualis arcui EA; </s>
  <s xml:id="echoid-s11948" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s11949" xml:space="preserve">concipiatur curva OGO ta-<lb/>lis, ut projectâ quâcunque rectâ DOL (quæ curvam OGO ſecet <lb/>puncto O, rectam ET in L) ductâque OQ ad GF parallelâ, ſit <lb/>VL. </s>
  <s xml:id="echoid-s11950" xml:space="preserve">AQ:</s>
  <s xml:id="echoid-s11951" xml:space="preserve">: X. </s>
  <s xml:id="echoid-s11952" xml:space="preserve">Y; </s>
  <s xml:id="echoid-s11953" xml:space="preserve">eſtque curva OGO (è ſuprà monſtratis) _Hy-_ <lb/>_perboln;_ </s>
  <s xml:id="echoid-s11954" xml:space="preserve">hanc tangat recta GS; </s>
  <s xml:id="echoid-s11955" xml:space="preserve">etiam recta GS curvam AGB <lb/>continget.</s>
  <s xml:id="echoid-s11956" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11957" xml:space="preserve">Nam concipiatur altera curva NGN talis, ut cùm hanc ſecet recta <lb/>arbitraria DL in N, curvam AEH in K, rectam TE in L; </s>
  <s xml:id="echoid-s11958" xml:space="preserve">ductáq; <lb/></s>
  <s xml:id="echoid-s11959" xml:space="preserve">ſit NR ad GF parallela, ſit VL + LK. </s>
  <s xml:id="echoid-s11960" xml:space="preserve">AR:</s>
  <s xml:id="echoid-s11961" xml:space="preserve">: X. </s>
  <s xml:id="echoid-s11962" xml:space="preserve">Y; </s>
  <s xml:id="echoid-s11963" xml:space="preserve">manife-<lb/>ſtum eſt curvam NGN utramque curvam AGB, &amp; </s>
  <s xml:id="echoid-s11964" xml:space="preserve">OGO tange-<lb/>re. </s>
  <s xml:id="echoid-s11965" xml:space="preserve">[ſecet enim recta DL curvam AEB in I, ducatúrque IP ad <lb/>GF parallela; </s>
  <s xml:id="echoid-s11966" xml:space="preserve">quum ergò ſit VL + LK. </s>
  <s xml:id="echoid-s11967" xml:space="preserve">AR:</s>
  <s xml:id="echoid-s11968" xml:space="preserve">: X. </s>
  <s xml:id="echoid-s11969" xml:space="preserve">Y:</s>
  <s xml:id="echoid-s11970" xml:space="preserve">: AK. </s>
  <s xml:id="echoid-s11971" xml:space="preserve"><lb/>AP, &amp; </s>
  <s xml:id="echoid-s11972" xml:space="preserve">ſit VL + LK &amp;</s>
  <s xml:id="echoid-s11973" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11974" xml:space="preserve">AK; </s>
  <s xml:id="echoid-s11975" xml:space="preserve">erit AR &amp;</s>
  <s xml:id="echoid-s11976" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s11977" xml:space="preserve">AP; </s>
  <s xml:id="echoid-s11978" xml:space="preserve">vel DR &amp;</s>
  <s xml:id="echoid-s11979" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11980" xml:space="preserve"><lb/>DP; </s>
  <s xml:id="echoid-s11981" xml:space="preserve">adeóque DN &amp;</s>
  <s xml:id="echoid-s11982" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s11983" xml:space="preserve">DI; </s>
  <s xml:id="echoid-s11984" xml:space="preserve">unde punctum N intra curvam AGB <lb/>ſemper cadet; </s>
  <s xml:id="echoid-s11985" xml:space="preserve">ac proinde curva NGN curvam AGB tan-<lb/>get; </s>
  <s xml:id="echoid-s11986" xml:space="preserve">ſimilique planè diſcurſu curva NGN curvam OGO contin-<lb/>get.</s>
  <s xml:id="echoid-s11987" xml:space="preserve">] Itaque curvæ AGB, OGO ſeſe (æquipollentèr) tangunt. </s>
  <s xml:id="echoid-s11988" xml:space="preserve"><lb/>Quare cùm recta GS curvam OGO tangat; </s>
  <s xml:id="echoid-s11989" xml:space="preserve">eadem curvam AGB <lb/>quoque continget: </s>
  <s xml:id="echoid-s11990" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s11991" xml:space="preserve">E. </s>
  <s xml:id="echoid-s11992" xml:space="preserve">F.</s>
  <s xml:id="echoid-s11993" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s11994" xml:space="preserve">Si curva AEH ſit circuli quadrans, cujus centrum D; </s>
  <s xml:id="echoid-s11995" xml:space="preserve">erit curva <lb/>AGB _Quadratrix communis_. </s>
  <s xml:id="echoid-s11996" xml:space="preserve">Ejus igitur _Tangens_ (unà cùm omni-<lb/>um ſimili ratione genitarum tangentibus) hoc pacto deſignatur,</s>
</p>
<pb o="78" file="0256" n="271" rhead=""/>
<p>
  <s xml:id="echoid-s11997" xml:space="preserve">Hujuſmodi plura quædam cogitaram hîc inſerere; </s>
  <s xml:id="echoid-s11998" xml:space="preserve">verùm hæc ex-<lb/>iſtimo ſufficere ſubindicando modo, juxta quem, citra _Calculi moleſti-_ <lb/>_am, curvarum tangentes_ exquirere licet, unáque conſtructiones de-<lb/>monſtrare. </s>
  <s xml:id="echoid-s11999" xml:space="preserve">Subjiciam tamen unum aut alterum non aſpernanda, ut vi-<lb/>detur _Theoremata_ perquam generalia.</s>
  <s xml:id="echoid-s12000" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12001" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s12002" xml:space="preserve">Sit linea quæpiam ZGE, cujus axis VD; </s>
  <s xml:id="echoid-s12003" xml:space="preserve">ad quam impri-<lb/>mìs applicatæ perpendiculares (VZ, PG, DE) ab initio VZ con-<lb/>
<anchor type="note" xlink:label="note-0256-01a" xlink:href="note-0256-01"/>
tinuè utcunque creſcant; </s>
  <s xml:id="echoid-s12004" xml:space="preserve">ſit item linea VIF talis, ut ductâ quâcunq; <lb/></s>
  <s xml:id="echoid-s12005" xml:space="preserve">rectâ EDF ad VD perpendiculari (quæ _curvas_ ſecet punctis E, F, <lb/>ipſam VD in D) ſit ſemper _rectangulum_ ex DF, &amp; </s>
  <s xml:id="echoid-s12006" xml:space="preserve">deſignatâ quâ-<lb/>dam R æquale _ſpatio_ reſpectivè _intercepto_ VDEZ; </s>
  <s xml:id="echoid-s12007" xml:space="preserve">fiat autem DE. </s>
  <s xml:id="echoid-s12008" xml:space="preserve"><lb/>DF:</s>
  <s xml:id="echoid-s12009" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s12010" xml:space="preserve">DT; </s>
  <s xml:id="echoid-s12011" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12012" xml:space="preserve">connectatur recta TF; </s>
  <s xml:id="echoid-s12013" xml:space="preserve">hæc curvam VIF <lb/>continget.</s>
  <s xml:id="echoid-s12014" xml:space="preserve"/>
</p>
<div xml:id="echoid-div361" type="float" level="2" n="12">
<note position="left" xlink:label="note-0256-01" xlink:href="note-0256-01a" xml:space="preserve">Fig. 109.</note>
</div>
<p>
  <s xml:id="echoid-s12015" xml:space="preserve">Sumatur enim in linea VIF punctum quodpiam I (illud primò ſu-<lb/>
<anchor type="note" xlink:label="note-0256-02a" xlink:href="note-0256-02"/>
pra punctum F, verſus initium V) &amp; </s>
  <s xml:id="echoid-s12016" xml:space="preserve">per hoc ducantur rectæ IG ad <lb/>VZ, ac KL ad VD parallelæ (quæ lineas expoſitas ſecent, ut vides) <lb/>éſtque tum LF. </s>
  <s xml:id="echoid-s12017" xml:space="preserve">LK:</s>
  <s xml:id="echoid-s12018" xml:space="preserve">: (DF. </s>
  <s xml:id="echoid-s12019" xml:space="preserve">DT:</s>
  <s xml:id="echoid-s12020" xml:space="preserve">:) DE. </s>
  <s xml:id="echoid-s12021" xml:space="preserve">R; </s>
  <s xml:id="echoid-s12022" xml:space="preserve">adeóque LF x <lb/>R = LK x DE. </s>
  <s xml:id="echoid-s12023" xml:space="preserve">Eſt autem (ex præſtituta linearum iſtarum natura) <lb/>LF x R æquale ſpatio PDEG; </s>
  <s xml:id="echoid-s12024" xml:space="preserve">ergò LK x DE = PDEG &amp;</s>
  <s xml:id="echoid-s12025" xml:space="preserve">lt; <lb/></s>
  <s xml:id="echoid-s12026" xml:space="preserve">DP x DE. </s>
  <s xml:id="echoid-s12027" xml:space="preserve">Unde eſt LK &amp;</s>
  <s xml:id="echoid-s12028" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s12029" xml:space="preserve">DP; </s>
  <s xml:id="echoid-s12030" xml:space="preserve">vel LK &amp;</s>
  <s xml:id="echoid-s12031" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s12032" xml:space="preserve">LI.</s>
  <s xml:id="echoid-s12033" xml:space="preserve"/>
</p>
<div xml:id="echoid-div362" type="float" level="2" n="13">
<note position="left" xlink:label="note-0256-02" xlink:href="note-0256-02a" xml:space="preserve">Fig. 110.</note>
</div>
<p>
  <s xml:id="echoid-s12034" xml:space="preserve">Rurſus accipiatur quodvis punctum I, infra punctum F, reliquáq; <lb/></s>
  <s xml:id="echoid-s12035" xml:space="preserve">fiant, utì priùs; </s>
  <s xml:id="echoid-s12036" xml:space="preserve">ſimilíque jam planè diſcurſu conſtabit fore LK x DE <lb/> = PDEG &amp;</s>
  <s xml:id="echoid-s12037" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s12038" xml:space="preserve">DP x DE, unde jam erit LK &amp;</s>
  <s xml:id="echoid-s12039" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s12040" xml:space="preserve">DP, vel LI. </s>
  <s xml:id="echoid-s12041" xml:space="preserve">E <lb/>quibus liquidò patet totam rectam TKFK intra (ſeu extra) curvam <lb/>VIFI exiſtere.</s>
  <s xml:id="echoid-s12042" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12043" xml:space="preserve">Iiſdem quoad cætera poſitis, ſi _ordinatæ_ VZ, PG, DE, &amp;</s>
  <s xml:id="echoid-s12044" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12045" xml:space="preserve">con-<lb/>tinuè decreſcant, eadem concluſio ſimili ratiocinio colligetur; </s>
  <s xml:id="echoid-s12046" xml:space="preserve">uni-<lb/>cum obvenit _Diſcrimen_, quòd in hoc caſu (contra quàm in priore) <lb/>linea VIF concavas ſu<unsure/>as axi VD obvertat.</s>
  <s xml:id="echoid-s12047" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12048" xml:space="preserve">_Corol_. </s>
  <s xml:id="echoid-s12049" xml:space="preserve">Notetur DE x DT æquari ſpatio VDEZ.</s>
  <s xml:id="echoid-s12050" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12051" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s12052" xml:space="preserve">Exindè deducitur hoc _Tbeorema_: </s>
  <s xml:id="echoid-s12053" xml:space="preserve">Sint duæ lineæ quævis <lb/>ZGE, VKF ta relatæ, ut ad communem ipſarum axem VD ap-<lb/>
<anchor type="note" xlink:label="note-0256-03a" xlink:href="note-0256-03"/>
plicatâ quâvis rectâ; </s>
  <s xml:id="echoid-s12054" xml:space="preserve">EDF, ſit ſemper quadratum ex DE æquale _du-_ <lb/>_plo ſpatio_ VDEZ; </s>
  <s xml:id="echoid-s12055" xml:space="preserve">ſumatur autem DQ = DE, &amp; </s>
  <s xml:id="echoid-s12056" xml:space="preserve">connectatur FQ; <lb/></s>
  <s xml:id="echoid-s12057" xml:space="preserve">hæc curvæ VKF perpendicularis erit.</s>
  <s xml:id="echoid-s12058" xml:space="preserve"/>
</p>
<div xml:id="echoid-div363" type="float" level="2" n="14">
<note position="left" xlink:label="note-0256-03" xlink:href="note-0256-03a" xml:space="preserve">Fig. 111.</note>
</div>
<p>
  <s xml:id="echoid-s12059" xml:space="preserve">Concipiatur enim linea VIF, per F tranſiens, talis qualem mox <lb/>attigimus (cujus ſcilicet ad VD applicatæ ſe habeant ut ſpatia VDEZ; <lb/></s>
  <s xml:id="echoid-s12060" xml:space="preserve">hoc eſt ut quadrata ex applicatis à curva VKF in præſente hypotheſi)
<pb o="79" file="0257" n="272" rhead=""/>
lineámque VIF tangat recta FT; </s>
  <s xml:id="echoid-s12061" xml:space="preserve">item lineam VKF tângat recta <lb/>
<anchor type="note" xlink:label="note-0257-01a" xlink:href="note-0257-01"/>
FS. </s>
  <s xml:id="echoid-s12062" xml:space="preserve">Eſt ergò SD <anchor type="note" xlink:href="" symbol="(_a_)"/> = 2 TD. </s>
  <s xml:id="echoid-s12063" xml:space="preserve">atqui DE x DT <anchor type="note" xlink:href="" symbol="(_b_)"/> = VDEZ.</s>
  <s xml:id="echoid-s12064" xml:space="preserve">
ergò DE x SD = (2 VDEZ = ) FDq. </s>
  <s xml:id="echoid-s12065" xml:space="preserve">unde conſtat angulum <lb/>
<anchor type="note" xlink:label="note-0257-02a" xlink:href="note-0257-02"/>
QFS rectum eſſe. </s>
  <s xml:id="echoid-s12066" xml:space="preserve">quod Propoſitum erat.</s>
  <s xml:id="echoid-s12067" xml:space="preserve"/>
</p>
<div xml:id="echoid-div364" type="float" level="2" n="15">
<note symbol="(_a_)" position="right" xlink:label="note-0257-01" xlink:href="note-0257-01a" xml:space="preserve">5. Lect. <lb/>IX.</note>
<note symbol="(_b_) it" position="right" xlink:label="note-0257-02" xlink:href="note-0257-02a" xml:space="preserve">Cor. præc.</note>
</div>
<p>
  <s xml:id="echoid-s12068" xml:space="preserve">Adjungam &amp; </s>
  <s xml:id="echoid-s12069" xml:space="preserve">illis cognata hæc.</s>
  <s xml:id="echoid-s12070" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12071" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s12072" xml:space="preserve">Sit curva quævis AGEZ, punctúmque quoddam D (à quo <lb/>projectæ DA, DG, DE, &amp; </s>
  <s xml:id="echoid-s12073" xml:space="preserve">_c_. </s>
  <s xml:id="echoid-s12074" xml:space="preserve">ab initio DA continuò decreſcant) <lb/>
<anchor type="note" xlink:label="note-0257-03a" xlink:href="note-0257-03"/>
tum altera ſit curva DKE, priorem interſecans in E, naturâque ta-<lb/>lis, ut à D utcunque projectâ rectâ DKG (quæ curvam AEZ ſecet <lb/>in G, curvam DKE in K) ſit perpetuò rectangulum ex DK, &amp; </s>
  <s xml:id="echoid-s12075" xml:space="preserve">de-<lb/>ſignatâ quâdam lineâ R æquale ſpatio ADG; </s>
  <s xml:id="echoid-s12076" xml:space="preserve">tum ductâ DT ad <lb/>DE perpendiculari, ſit DT = 2 R; </s>
  <s xml:id="echoid-s12077" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12078" xml:space="preserve">connectatur TE; </s>
  <s xml:id="echoid-s12079" xml:space="preserve">hæc <lb/>curvam DKE continget.</s>
  <s xml:id="echoid-s12080" xml:space="preserve"/>
</p>
<div xml:id="echoid-div365" type="float" level="2" n="16">
<note position="right" xlink:label="note-0257-03" xlink:href="note-0257-03a" xml:space="preserve">Fig. 112.</note>
</div>
<p>
  <s xml:id="echoid-s12081" xml:space="preserve">Nam ſumpto quovis in curva DKE puncto K, ducatur recta DKG; <lb/></s>
  <s xml:id="echoid-s12082" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12083" xml:space="preserve">ſumptâ DL = DK, ducatur LR ad DT parallela ( ſecans ipſam <lb/>DG in Y). </s>
  <s xml:id="echoid-s12084" xml:space="preserve">tum per E ducatur EX ad DE perpendicularis (hæc <lb/>verò extra curvam AEZ, ad partes Z cadet, quia decreſcunt proje-<lb/>ctæ verſus Z; </s>
  <s xml:id="echoid-s12085" xml:space="preserve">unde EX verſus A intra curvam EGA cadet; </s>
  <s xml:id="echoid-s12086" xml:space="preserve">eate-<lb/>
<anchor type="note" xlink:label="note-0257-04a" xlink:href="note-0257-04"/>
nus ſaltem, quatenus huic Propoſito ſatisfaciet). </s>
  <s xml:id="echoid-s12087" xml:space="preserve">Sit jam primò pun-<lb/>ctum G ſupra E, verſus initium A, &amp; </s>
  <s xml:id="echoid-s12088" xml:space="preserve">ob TD. </s>
  <s xml:id="echoid-s12089" xml:space="preserve">DE:</s>
  <s xml:id="echoid-s12090" xml:space="preserve">: RL. </s>
  <s xml:id="echoid-s12091" xml:space="preserve">LE; <lb/></s>
  <s xml:id="echoid-s12092" xml:space="preserve">
<anchor type="note" xlink:label="note-0257-05a" xlink:href="note-0257-05"/>
adeóque RL x DE = TD x LE (a) = 2 R x LE (a) = 2 GDE <lb/>&amp;</s>
  <s xml:id="echoid-s12093" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s12094" xml:space="preserve">2 DEX = EX x DE. </s>
  <s xml:id="echoid-s12095" xml:space="preserve">ergò RL &amp;</s>
  <s xml:id="echoid-s12096" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s12097" xml:space="preserve">EX &amp;</s>
  <s xml:id="echoid-s12098" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s12099" xml:space="preserve">LY. </s>
  <s xml:id="echoid-s12100" xml:space="preserve">Eſt autem <lb/>punctum Y extra curvam, quia DY &amp;</s>
  <s xml:id="echoid-s12101" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s12102" xml:space="preserve">DL = DK; </s>
  <s xml:id="echoid-s12103" xml:space="preserve">ergò magìs <lb/>punctum R eſt extra curvam.</s>
  <s xml:id="echoid-s12104" xml:space="preserve"/>
</p>
<div xml:id="echoid-div366" type="float" level="2" n="17">
<note position="right" xlink:label="note-0257-04" xlink:href="note-0257-04a" xml:space="preserve">Fig. 113.</note>
<note symbol="(_a_) it" position="right" xlink:label="note-0257-05" xlink:href="note-0257-05a" xml:space="preserve">Hyp.</note>
</div>
<p>
  <s xml:id="echoid-s12105" xml:space="preserve">Sit rurſus punctum G infra punctum E verſus Z; </s>
  <s xml:id="echoid-s12106" xml:space="preserve">eſtque rurſus, <lb/>utì priùs, RL x DE = 2 GDE &amp;</s>
  <s xml:id="echoid-s12107" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s12108" xml:space="preserve">2 triang. </s>
  <s xml:id="echoid-s12109" xml:space="preserve">EDX = EX x DE. <lb/></s>
  <s xml:id="echoid-s12110" xml:space="preserve">unde RL &amp;</s>
  <s xml:id="echoid-s12111" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s12112" xml:space="preserve">EX &amp;</s>
  <s xml:id="echoid-s12113" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s12114" xml:space="preserve">LY. </s>
  <s xml:id="echoid-s12115" xml:space="preserve">Eſt autem recta LY extra curvam EK <lb/>tota, (nam etiam extra arcum LK curvæ KE circumductum tota ja-<lb/>cet) ergò punctum R rurſus extra curvam exiſtit. </s>
  <s xml:id="echoid-s12116" xml:space="preserve">Liquidum eſt igi-<lb/>tur rectam TER curvam DKE tangere.</s>
  <s xml:id="echoid-s12117" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12118" xml:space="preserve">Quòd ſi punctum aliud ìn curva DKE deſignetur, puta K; </s>
  <s xml:id="echoid-s12119" xml:space="preserve">per <lb/>quod ducta ſit DKG; </s>
  <s xml:id="echoid-s12120" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12121" xml:space="preserve">fiat DG. </s>
  <s xml:id="echoid-s12122" xml:space="preserve">DK:</s>
  <s xml:id="echoid-s12123" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s12124" xml:space="preserve">P; </s>
  <s xml:id="echoid-s12125" xml:space="preserve">ſumatúrque <lb/>DT = 2 P; </s>
  <s xml:id="echoid-s12126" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12127" xml:space="preserve">connectatur TG; </s>
  <s xml:id="echoid-s12128" xml:space="preserve">tum ducatur KS ad GT paralle-<lb/>la; </s>
  <s xml:id="echoid-s12129" xml:space="preserve">recta KS curvam DKE tanget.</s>
  <s xml:id="echoid-s12130" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12131" xml:space="preserve">Nam concipiatur curva DOG, per G tranſiens, talis, ut rectâ <lb/>quâcunque DON à D projectâ (quæ curvam DOG ſecet in O, <lb/>curvam DNE in M, curvam AGE in N) ſit ſemper DO x P æ-<lb/>qualis ſpatio ADN; </s>
  <s xml:id="echoid-s12132" xml:space="preserve">erit ideò DM x R = DO x P; </s>
  <s xml:id="echoid-s12133" xml:space="preserve">ac proinde <lb/>DM. </s>
  <s xml:id="echoid-s12134" xml:space="preserve">DO:</s>
  <s xml:id="echoid-s12135" xml:space="preserve">: P. </s>
  <s xml:id="echoid-s12136" xml:space="preserve">R. </s>
  <s xml:id="echoid-s12137" xml:space="preserve">unde lì<unsure/>neæ DKE, DOG analogæ erunt. </s>
  <s xml:id="echoid-s12138" xml:space="preserve">Ve-
<pb o="80" file="0258" n="273" rhead=""/>
rùm ex jam modò oſtenſis GT curvam DOG tangit; </s>
  <s xml:id="echoid-s12139" xml:space="preserve">ergò KS ip-<lb/>ſam DKE continget.</s>
  <s xml:id="echoid-s12140" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12141" xml:space="preserve">Notetur eſſe DG q. </s>
  <s xml:id="echoid-s12142" xml:space="preserve">DK q:</s>
  <s xml:id="echoid-s12143" xml:space="preserve">: 2 R. </s>
  <s xml:id="echoid-s12144" xml:space="preserve">DS.</s>
  <s xml:id="echoid-s12145" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12146" xml:space="preserve">Nam eſt DG q. </s>
  <s xml:id="echoid-s12147" xml:space="preserve">DK q = DG. </s>
  <s xml:id="echoid-s12148" xml:space="preserve">DK + DG. </s>
  <s xml:id="echoid-s12149" xml:space="preserve">DK = R. </s>
  <s xml:id="echoid-s12150" xml:space="preserve">P + <lb/>DT. </s>
  <s xml:id="echoid-s12151" xml:space="preserve">DS = R. </s>
  <s xml:id="echoid-s12152" xml:space="preserve">P + 2 P. </s>
  <s xml:id="echoid-s12153" xml:space="preserve">DS = 2 RP. </s>
  <s xml:id="echoid-s12154" xml:space="preserve">P x DS = 2 R. </s>
  <s xml:id="echoid-s12155" xml:space="preserve">DS. <lb/></s>
  <s xml:id="echoid-s12156" xml:space="preserve">itaque DG q. </s>
  <s xml:id="echoid-s12157" xml:space="preserve">DKQ:</s>
  <s xml:id="echoid-s12158" xml:space="preserve">: 2 R. </s>
  <s xml:id="echoid-s12159" xml:space="preserve">DS.</s>
  <s xml:id="echoid-s12160" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12161" xml:space="preserve">Hæc autem perinde vera ſunt, nec abſimili modo demonſtrantur; <lb/></s>
  <s xml:id="echoid-s12162" xml:space="preserve">etiam ſi projectæ à D rectæ DA, DG, DE, &amp;_</s>
  <s xml:id="echoid-s12163" xml:space="preserve">c_. </s>
  <s xml:id="echoid-s12164" xml:space="preserve">pares ſint (quo ca-<lb/>ſu curva AGEZ _Circulus_ erit, &amp; </s>
  <s xml:id="echoid-s12165" xml:space="preserve">_Curva_ DKE _Spiralis Archimedæa_) <lb/>aut à DA continuò creſcant.</s>
  <s xml:id="echoid-s12166" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12167" xml:space="preserve">Exindè verò facilè colligitur hoc _Theorema_:</s>
  <s xml:id="echoid-s12168" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12169" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s12170" xml:space="preserve">Sint duæ curvæ AGE, DKE ità verſus ſe relatæ, ut à de-<lb/>ſignato in curva DKE puncto D ductis rectis DA, DG (quarum <lb/>hæc ipſam DKE ſecetin K) ſit ſemper _Quadratum_ ex DK _Quadru-_ <lb/>
<anchor type="note" xlink:label="note-0258-01a" xlink:href="note-0258-01"/>
_plum ſpatii_ ADG; </s>
  <s xml:id="echoid-s12171" xml:space="preserve">ductâ DH ad DG perpendiculari, &amp; </s>
  <s xml:id="echoid-s12172" xml:space="preserve">facto DK. <lb/></s>
  <s xml:id="echoid-s12173" xml:space="preserve">DG:</s>
  <s xml:id="echoid-s12174" xml:space="preserve">: DG. </s>
  <s xml:id="echoid-s12175" xml:space="preserve">DH; </s>
  <s xml:id="echoid-s12176" xml:space="preserve">connexâque HK; </s>
  <s xml:id="echoid-s12177" xml:space="preserve">erit HK curvæ DKE per-<lb/>pendicularis.</s>
  <s xml:id="echoid-s12178" xml:space="preserve"/>
</p>
<div xml:id="echoid-div367" type="float" level="2" n="18">
<note position="left" xlink:label="note-0258-01" xlink:href="note-0258-01a" xml:space="preserve">Fig. 114.</note>
</div>
<p>
  <s xml:id="echoid-s12179" xml:space="preserve">Nam concipiatur linea DOKO, per K tranſiens, naturâque talis <lb/>ut ad illam à D projectæ (ceu DK) ſe habeant in eadem quâ ſpatia ADG <lb/>ratione (quales lineas attigimus in proximè ſuperiori) &amp; </s>
  <s xml:id="echoid-s12180" xml:space="preserve">lineam <lb/>DOK tangat recta KT, lineam DKE recta KS; </s>
  <s xml:id="echoid-s12181" xml:space="preserve">conveniant âu-<lb/>tem hæ cum ipſa HD punctis T, S; </s>
  <s xml:id="echoid-s12182" xml:space="preserve">eſt igitur (è præcedente) DG@q. <lb/></s>
  <s xml:id="echoid-s12183" xml:space="preserve">DKq:</s>
  <s xml:id="echoid-s12184" xml:space="preserve">: {DK/2}. </s>
  <s xml:id="echoid-s12185" xml:space="preserve">DT. </s>
  <s xml:id="echoid-s12186" xml:space="preserve">hoc eft DH. </s>
  <s xml:id="echoid-s12187" xml:space="preserve">DK:</s>
  <s xml:id="echoid-s12188" xml:space="preserve">: {DK.</s>
  <s xml:id="echoid-s12189" xml:space="preserve">/2} DT; </s>
  <s xml:id="echoid-s12190" xml:space="preserve">hoc eſt (quo-<lb/>niam è <anchor type="note" xlink:href="" symbol="*"/> mox præmonſtratis DS = 2 DT) DH. </s>
  <s xml:id="echoid-s12191" xml:space="preserve">DK:</s>
  <s xml:id="echoid-s12192" xml:space="preserve">: ({DK.</s>
  <s xml:id="echoid-s12193" xml:space="preserve">/2} {DS.</s>
  <s xml:id="echoid-s12194" xml:space="preserve">/2}
<anchor type="note" xlink:label="note-0258-02a" xlink:href="note-0258-02"/>
:</s>
  <s xml:id="echoid-s12195" xml:space="preserve">:) DK. </s>
  <s xml:id="echoid-s12196" xml:space="preserve">DS. </s>
  <s xml:id="echoid-s12197" xml:space="preserve">Liquet igitur rectam HK tangenti KS perpendicu-<lb/>larem eſſe: </s>
  <s xml:id="echoid-s12198" xml:space="preserve">Q. </s>
  <s xml:id="echoid-s12199" xml:space="preserve">E. </s>
  <s xml:id="echoid-s12200" xml:space="preserve">D.</s>
  <s xml:id="echoid-s12201" xml:space="preserve"/>
</p>
<div xml:id="echoid-div368" type="float" level="2" n="19">
<note symbol="* it" position="left" xlink:label="note-0258-02" xlink:href="note-0258-02a" xml:space="preserve">In 12 hujus.</note>
</div>
<p>
  <s xml:id="echoid-s12202" xml:space="preserve">Ità Propoſiti noſtri priore (quam innuebamus) parte quomodo-<lb/>t<unsure/>unque defuncti ſumus. </s>
  <s xml:id="echoid-s12203" xml:space="preserve">Cui ſupplendæ, appendiculæ inſtar, ſub-<lb/>nectemus à nobis uſitatum methodum ex Calculo tangentes reperien-<lb/>di. </s>
  <s xml:id="echoid-s12204" xml:space="preserve">Quanquam haud ſcio, poſt tot ejuſmodi pervulgatas atque pro-<lb/>tritas methodos, an id ex uſu ſit facere. </s>
  <s xml:id="echoid-s12205" xml:space="preserve">Facio ſaltem ex Amici con-<lb/>ſilio; </s>
  <s xml:id="echoid-s12206" xml:space="preserve">eóque libentiùs, quòd præ cæteris, quas tractavi, compendio-<lb/>ſa videtur, ac generalis. </s>
  <s xml:id="echoid-s12207" xml:space="preserve">In hunc procedo modum.</s>
  <s xml:id="echoid-s12208" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12209" xml:space="preserve">Sint AP, PM poſitione datæ rectæ lineæ (quarum PM propo-<lb/>ſitam curvam ſecet in M) &amp; </s>
  <s xml:id="echoid-s12210" xml:space="preserve">MT curvam tangere ponatur ad M,
<pb o="81" file="0259" n="274" rhead=""/>
rectam AP ſecare ad T; </s>
  <s xml:id="echoid-s12211" xml:space="preserve">ut ipſius jam rectæ PT quantitatem exqui-<lb/>
<anchor type="note" xlink:label="note-0259-01a" xlink:href="note-0259-01"/>
ram; </s>
  <s xml:id="echoid-s12212" xml:space="preserve">curvæ arcum MN indefinitè parvum ſtatuo; </s>
  <s xml:id="echoid-s12213" xml:space="preserve">tum duco rectas <lb/>NQ ad MP, &amp; </s>
  <s xml:id="echoid-s12214" xml:space="preserve">NR ad AP parallelas; </s>
  <s xml:id="echoid-s12215" xml:space="preserve">nomino MP = _m_; </s>
  <s xml:id="echoid-s12216" xml:space="preserve">PT <lb/> = _t_; </s>
  <s xml:id="echoid-s12217" xml:space="preserve">MR = _a_; </s>
  <s xml:id="echoid-s12218" xml:space="preserve">NR = _e_; </s>
  <s xml:id="echoid-s12219" xml:space="preserve">reliquáſque rectas, ex ſpeciali curvæ <lb/>natura determinatas, utiles propoſito, nominibus deſigno; </s>
  <s xml:id="echoid-s12220" xml:space="preserve">ipſas au-<lb/>tem MR, NR (&amp; </s>
  <s xml:id="echoid-s12221" xml:space="preserve">mediantibus illis ipſas MP, PT) per _æquationem_ <lb/>è Calculo deprehenſam inter ſe comparo; </s>
  <s xml:id="echoid-s12222" xml:space="preserve">regulas interim has obſer-<lb/>vans. </s>
  <s xml:id="echoid-s12223" xml:space="preserve">1. </s>
  <s xml:id="echoid-s12224" xml:space="preserve">Inter computandum omnes abjicio terminos, in quibus <lb/>ipſarum _a_, vel _e_ poteſtas habetur, vel in quibus ipſæ ducuntur in ſe <lb/>(etenim iſti termini nihil valebunt).</s>
  <s xml:id="echoid-s12225" xml:space="preserve"/>
</p>
<div xml:id="echoid-div369" type="float" level="2" n="20">
<note position="right" xlink:label="note-0259-01" xlink:href="note-0259-01a" xml:space="preserve">Fig. 115.</note>
</div>
<p>
  <s xml:id="echoid-s12226" xml:space="preserve">2. </s>
  <s xml:id="echoid-s12227" xml:space="preserve">Poſt _æquationem constitutam_, omnes abjicio terminos, literis <lb/>conftantes quantitates notas, ſeu determinatas deſignantibus; </s>
  <s xml:id="echoid-s12228" xml:space="preserve">aut in <lb/>quibus non habentur _a_, vel _e_. </s>
  <s xml:id="echoid-s12229" xml:space="preserve">(etenim illi termini ſemper, ad unam <lb/>æquationis partem adducti, nihilum adæquabunt).</s>
  <s xml:id="echoid-s12230" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12231" xml:space="preserve">3. </s>
  <s xml:id="echoid-s12232" xml:space="preserve">Pro _a_ ipſam _m_; </s>
  <s xml:id="echoid-s12233" xml:space="preserve">(vel MP) pro _e_ ipſam _t_ (vel PT) ſubſtituo. <lb/></s>
  <s xml:id="echoid-s12234" xml:space="preserve">Hinc demùm ipſius PT quantitas dignoſcetur.</s>
  <s xml:id="echoid-s12235" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12236" xml:space="preserve">Quòd ſi calculum ingrediatur curvæ cujuſpiam indefinita particula; <lb/></s>
  <s xml:id="echoid-s12237" xml:space="preserve">ſubſtituatur ejus loco tangentis particula ritè ſumpta; </s>
  <s xml:id="echoid-s12238" xml:space="preserve">vel ei quævis <lb/>(ob indefinitam curvæ parvitatem) æquipollens recta.</s>
  <s xml:id="echoid-s12239" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12240" xml:space="preserve">Hæc autem è ſubnexis Exemplis clariùs eluceſcent.</s>
  <s xml:id="echoid-s12241" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div371" type="section" level="1" n="36">
<head xml:id="echoid-head39" xml:space="preserve">Exemp. I.</head>
<p>
  <s xml:id="echoid-s12242" xml:space="preserve">Angulus ABH rectus ſit; </s>
  <s xml:id="echoid-s12243" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12244" xml:space="preserve">ſit curva AMO talis, ut per A du-<lb/>ctâ utcunque rectâ AK, quæ rectam BH ſecet in K, curvam AMO <lb/>
<anchor type="note" xlink:label="note-0259-02a" xlink:href="note-0259-02"/>
in M, ſit ſemper ſubtenſa AM æqualis abſciſſæ BK; </s>
  <s xml:id="echoid-s12245" xml:space="preserve">hujus curvæ ad <lb/>M tangens eſt deſignanda.</s>
  <s xml:id="echoid-s12246" xml:space="preserve"/>
</p>
<div xml:id="echoid-div371" type="float" level="2" n="1">
<note position="right" xlink:label="note-0259-02" xlink:href="note-0259-02a" xml:space="preserve">Fig. 116.</note>
</div>
<p>
  <s xml:id="echoid-s12247" xml:space="preserve">Fiant quæ ſuprà præſcripta ſunt, &amp; </s>
  <s xml:id="echoid-s12248" xml:space="preserve">(ductâ ANL) nominetur <lb/>AB = _r_; </s>
  <s xml:id="echoid-s12249" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12250" xml:space="preserve">AP = _q_; </s>
  <s xml:id="echoid-s12251" xml:space="preserve">unde AQ = _q_ - _e_; </s>
  <s xml:id="echoid-s12252" xml:space="preserve">item QN = _m_ -<lb/>_a_. </s>
  <s xml:id="echoid-s12253" xml:space="preserve">ergò eſt _qq_ + _ee_ - 2 _qe_ + _mm_ + _aa_ - 2 _ma_ = (AQq <lb/>+ QNq = ANq = ) BLq; </s>
  <s xml:id="echoid-s12254" xml:space="preserve">hoc eſt (rejectis, uti monitum eſt, <lb/>rejiciendis) _qq_ - 2 _qe_ + _mm_ - 2 _ma_ = BLq. </s>
  <s xml:id="echoid-s12255" xml:space="preserve">Porrò eſt <lb/>AQ. </s>
  <s xml:id="echoid-s12256" xml:space="preserve">QN:</s>
  <s xml:id="echoid-s12257" xml:space="preserve">: AB. </s>
  <s xml:id="echoid-s12258" xml:space="preserve">BL; </s>
  <s xml:id="echoid-s12259" xml:space="preserve">hoc eſt _q_ - _e. </s>
  <s xml:id="echoid-s12260" xml:space="preserve">m_ - _a_:</s>
  <s xml:id="echoid-s12261" xml:space="preserve">: _r._ </s>
  <s xml:id="echoid-s12262" xml:space="preserve">BL = <lb/>{_rm_ - _ra_.</s>
  <s xml:id="echoid-s12263" xml:space="preserve">/_q_ - _e_} quare {_rrmm_ + _rraa_ - 2 _rrma_/_qq_ + _ee_ - 2 _qe_.</s>
  <s xml:id="echoid-s12264" xml:space="preserve">} = BLq; </s>
  <s xml:id="echoid-s12265" xml:space="preserve">ſeu<unsure/> <lb/>(rejectis ſuperfluis) {_rrmm_ - 2 _rrma_/_qq_ - 2 _qe@_} = BLq = _qq_ - 2 _qe_ + <lb/>_mm_ - 2 _ma_. </s>
  <s xml:id="echoid-s12266" xml:space="preserve">vel _rrmm_ - 2 _rrma_ = _q_<emph style="sub">4</emph> - 2 _q_<emph style="sub">3</emph>_e_ + _qqmm_ - 2 _qqma_ - 2 _q_<emph style="sub">3</emph>_e_ + <lb/>4 _qqee_ - 2 _qmme_ + 4 _qmae_; </s>
  <s xml:id="echoid-s12267" xml:space="preserve">hoc eſt (abjectis iis, quæ præſcripſimus
<pb o="82" file="0260" n="275" rhead=""/>
abjicienda) - 2 _rrma_ = - 4 _q_<emph style="sub">3</emph>_e_ - 2 _qqma_ - 2 _qmme_. </s>
  <s xml:id="echoid-s12268" xml:space="preserve">vel <lb/>_rrma_ - qq_ma_ = 2 _q_<emph style="sub">3</emph>_e_ + _qmme_; </s>
  <s xml:id="echoid-s12269" xml:space="preserve">vel denuò ſubſtituendo _m_ <lb/>pro _a_, &amp; </s>
  <s xml:id="echoid-s12270" xml:space="preserve">_t_ pro _e_, eſt _rrmm_ - _qqmm_ = 2 _q_<emph style="sub">3</emph>_t_ - _qmmt_; </s>
  <s xml:id="echoid-s12271" xml:space="preserve">vel <lb/>{_rrmm_ - qq_mm_/2 q<emph style="sub">3</emph> - q_mm_} = _t_ = PT.</s>
  <s xml:id="echoid-s12272" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div373" type="section" level="1" n="37">
<head xml:id="echoid-head40" xml:space="preserve">_Exemp_. II.</head>
<p>
  <s xml:id="echoid-s12273" xml:space="preserve">Sit recta EA (poſitione ac magnitudine data) &amp; </s>
  <s xml:id="echoid-s12274" xml:space="preserve">curva EMO <lb/>proprietate talis, ut ab ea utcunque ductâ rectâ MP ad EA perpen-<lb/>
<anchor type="note" xlink:label="note-0260-01a" xlink:href="note-0260-01"/>
diculari _Summa Cuborum_ ex AP, &amp; </s>
  <s xml:id="echoid-s12275" xml:space="preserve">MP æquetur _Cubo_ rectæ AE.</s>
  <s xml:id="echoid-s12276" xml:space="preserve"/>
</p>
<div xml:id="echoid-div373" type="float" level="2" n="1">
<note position="left" xlink:label="note-0260-01" xlink:href="note-0260-01a" xml:space="preserve">Fig. 117.</note>
</div>
<p>
  <s xml:id="echoid-s12277" xml:space="preserve">Nominentur AE = _r_; </s>
  <s xml:id="echoid-s12278" xml:space="preserve">AP = _f_; </s>
  <s xml:id="echoid-s12279" xml:space="preserve">unde AQ = _f_ + _e_; </s>
  <s xml:id="echoid-s12280" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12281" xml:space="preserve">AQ <lb/>cub. </s>
  <s xml:id="echoid-s12282" xml:space="preserve">= _f_<emph style="sub">3</emph> + 3 _ffe_ + 3 _fee_ + _e_<emph style="sub">3</emph>; </s>
  <s xml:id="echoid-s12283" xml:space="preserve">(ſeu abjectis ſuperfluis, ex præ-<lb/>ſcripto) = _f_<emph style="sub">3</emph> + 3 _ffe_. </s>
  <s xml:id="echoid-s12284" xml:space="preserve">Item NQ cub. </s>
  <s xml:id="echoid-s12285" xml:space="preserve">= cub. </s>
  <s xml:id="echoid-s12286" xml:space="preserve">_m_ - _a_ = _m_<emph style="sub">3</emph> -<lb/>3 _mma_ + 3 _maa_ - _a_<emph style="sub">3</emph> (hoc eſt) = _m_<emph style="sub">3</emph> - 3 _mma_. </s>
  <s xml:id="echoid-s12287" xml:space="preserve">Quapropter <lb/>eſt _f_<emph style="sub">3</emph> + 3 _ffe_ + _m_<emph style="sub">3</emph> - 3 _mma_ = (AQ cub. </s>
  <s xml:id="echoid-s12288" xml:space="preserve">+ NQ cub. </s>
  <s xml:id="echoid-s12289" xml:space="preserve">= <lb/>AE cub. </s>
  <s xml:id="echoid-s12290" xml:space="preserve">= ) _r_<emph style="sub">3</emph>. </s>
  <s xml:id="echoid-s12291" xml:space="preserve">abjectíſque datis, eſt 3 _ffe_ = 3 _mma_ = _o_. <lb/></s>
  <s xml:id="echoid-s12292" xml:space="preserve">ſeu, _ffe_ = _mma_; </s>
  <s xml:id="echoid-s12293" xml:space="preserve">ſubrogatíſque loco _a_, &amp; </s>
  <s xml:id="echoid-s12294" xml:space="preserve">_e_ ipſis _m_, &amp; </s>
  <s xml:id="echoid-s12295" xml:space="preserve">_t_, erit <lb/>_fft_ = _m_<emph style="sub">3</emph>; </s>
  <s xml:id="echoid-s12296" xml:space="preserve">ſeu _t_ = {_m_<emph style="sub">3</emph>/_ff_}; </s>
  <s xml:id="echoid-s12297" xml:space="preserve">eſt ergò PT quarta proportionalis in ratio-<lb/>ne AP ad PM continuata.</s>
  <s xml:id="echoid-s12298" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12299" xml:space="preserve">Similiter, Si fuerit APqq + MPqq = AEqq; </s>
  <s xml:id="echoid-s12300" xml:space="preserve">reperietur <lb/>fore PT = {_m_<emph style="sub">4</emph>/_f_<emph style="sub">3</emph>}; </s>
  <s xml:id="echoid-s12301" xml:space="preserve">vel PM quarta proportionalis in ratione AP ad <lb/>PM; </s>
  <s xml:id="echoid-s12302" xml:space="preserve">ac ità porrò quod de _Cycloformibus_ iſtis lineis an obſervatu <lb/>dignum ſit neſcio.</s>
  <s xml:id="echoid-s12303" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div375" type="section" level="1" n="38">
<head xml:id="echoid-head41" xml:space="preserve">_Exemp_. III</head>
<p>
  <s xml:id="echoid-s12304" xml:space="preserve">Poſitione data ſit recta AZ, &amp; </s>
  <s xml:id="echoid-s12305" xml:space="preserve">AX magnitudine; </s>
  <s xml:id="echoid-s12306" xml:space="preserve">ſit etiam _curva_ <lb/>AMO talis, ut ductâ utcunque rectâ MP ad AZ normali, ſit AP <lb/>
<anchor type="note" xlink:label="note-0260-02a" xlink:href="note-0260-02"/>
_cub._ </s>
  <s xml:id="echoid-s12307" xml:space="preserve">+ PM _cub_. </s>
  <s xml:id="echoid-s12308" xml:space="preserve">= AX x AP x PM.</s>
  <s xml:id="echoid-s12309" xml:space="preserve"/>
</p>
<div xml:id="echoid-div375" type="float" level="2" n="1">
<note position="left" xlink:label="note-0260-02" xlink:href="note-0260-02a" xml:space="preserve">Fig. 118. <lb/>_La Galande_</note>
</div>
<p>
  <s xml:id="echoid-s12310" xml:space="preserve">Dicantur AX = _b_; </s>
  <s xml:id="echoid-s12311" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12312" xml:space="preserve">AP = _f_; </s>
  <s xml:id="echoid-s12313" xml:space="preserve">ergò AQ = _f_ - _e_; </s>
  <s xml:id="echoid-s12314" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12315" xml:space="preserve">AQ <lb/>_cub_. </s>
  <s xml:id="echoid-s12316" xml:space="preserve">= _f_<emph style="sub">3</emph> - 3 _ffe_; </s>
  <s xml:id="echoid-s12317" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12318" xml:space="preserve">QN _cub._ </s>
  <s xml:id="echoid-s12319" xml:space="preserve">= _m_<emph style="sub">3</emph> - 3 _mma_. </s>
  <s xml:id="echoid-s12320" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12321" xml:space="preserve">AQ x <lb/>QN = _fm_ - _fa_ - _me_ + _ae_ = _fm_ - _fa_ - _me_; </s>
  <s xml:id="echoid-s12322" xml:space="preserve">unde AX x <lb/>AQ x QN = _bfm_ - _bfa_ - _bme_; </s>
  <s xml:id="echoid-s12323" xml:space="preserve">hinc æquatio _f_ - 3 _ffe_ <lb/>+ _m_<emph style="sub">3</emph> - 3 _mma_ = _bfm_ - _bfa_ - _bme_; </s>
  <s xml:id="echoid-s12324" xml:space="preserve">ſeu amoliendo reje-
<pb o="83" file="0261" n="276" rhead=""/>
ctanea, _bfa_ - 3 _mma_ = 3 _ffe_ - _bme_; </s>
  <s xml:id="echoid-s12325" xml:space="preserve">ſubſtituendóque _bfm_ -<lb/>3 _m_<emph style="sub">3</emph> = 3 _fft_ - _bmt_; </s>
  <s xml:id="echoid-s12326" xml:space="preserve">ſeu, {_bfm_ - 3 _m_<emph style="sub">3</emph>/3 _ff_ - _bm_} = _t_.</s>
  <s xml:id="echoid-s12327" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div377" type="section" level="1" n="39">
<head xml:id="echoid-head42" xml:space="preserve">Exemp. IV.</head>
<p>
  <s xml:id="echoid-s12328" xml:space="preserve">Sit _Quadratrix_ CMV (ad circulum CEB pertinens cui centrum <lb/>A,) cujus axis VA; </s>
  <s xml:id="echoid-s12329" xml:space="preserve">ordinatæ CA. </s>
  <s xml:id="echoid-s12330" xml:space="preserve">MP ad VA perpendicula-<lb/>res.</s>
  <s xml:id="echoid-s12331" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12332" xml:space="preserve">Protractis rectis AME, ANF, ductíſque rectis EK, FL ad AB <lb/>
<anchor type="note" xlink:label="note-0261-01a" xlink:href="note-0261-01"/>
perpendicularibus, dicantur arcus CB = _p_; </s>
  <s xml:id="echoid-s12333" xml:space="preserve">radius AC = _r_; </s>
  <s xml:id="echoid-s12334" xml:space="preserve">recta <lb/>AP = _f_; </s>
  <s xml:id="echoid-s12335" xml:space="preserve">AM = _k_. </s>
  <s xml:id="echoid-s12336" xml:space="preserve">Eſtque jam CA arc. </s>
  <s xml:id="echoid-s12337" xml:space="preserve">CB:</s>
  <s xml:id="echoid-s12338" xml:space="preserve">: NR. </s>
  <s xml:id="echoid-s12339" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s12340" xml:space="preserve">FE. <lb/></s>
  <s xml:id="echoid-s12341" xml:space="preserve">hoc eſt, _r. </s>
  <s xml:id="echoid-s12342" xml:space="preserve">p_:</s>
  <s xml:id="echoid-s12343" xml:space="preserve">: _a_. </s>
  <s xml:id="echoid-s12344" xml:space="preserve">{_pa_/_r_} = arc. </s>
  <s xml:id="echoid-s12345" xml:space="preserve">FE. </s>
  <s xml:id="echoid-s12346" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12347" xml:space="preserve">AM. </s>
  <s xml:id="echoid-s12348" xml:space="preserve">MP:</s>
  <s xml:id="echoid-s12349" xml:space="preserve">: AE. </s>
  <s xml:id="echoid-s12350" xml:space="preserve">EK; </s>
  <s xml:id="echoid-s12351" xml:space="preserve">hoc <lb/>eſt, _k. </s>
  <s xml:id="echoid-s12352" xml:space="preserve">m_:</s>
  <s xml:id="echoid-s12353" xml:space="preserve">: _r_. </s>
  <s xml:id="echoid-s12354" xml:space="preserve">{_rm_/_k_} = EK; </s>
  <s xml:id="echoid-s12355" xml:space="preserve">item AE. </s>
  <s xml:id="echoid-s12356" xml:space="preserve">EK:</s>
  <s xml:id="echoid-s12357" xml:space="preserve">: arc. </s>
  <s xml:id="echoid-s12358" xml:space="preserve">FE. </s>
  <s xml:id="echoid-s12359" xml:space="preserve">LK. </s>
  <s xml:id="echoid-s12360" xml:space="preserve">hoc <lb/>eſt, _r_. </s>
  <s xml:id="echoid-s12361" xml:space="preserve">{_rm_:</s>
  <s xml:id="echoid-s12362" xml:space="preserve">: _pa_/_kr_.</s>
  <s xml:id="echoid-s12363" xml:space="preserve">} {_pma_/_rk_} = LK. </s>
  <s xml:id="echoid-s12364" xml:space="preserve">Verùm AM. </s>
  <s xml:id="echoid-s12365" xml:space="preserve">AE:</s>
  <s xml:id="echoid-s12366" xml:space="preserve">: AP. </s>
  <s xml:id="echoid-s12367" xml:space="preserve">AK; </s>
  <s xml:id="echoid-s12368" xml:space="preserve"><lb/>hoc eſt _k_._</s>
  <s xml:id="echoid-s12369" xml:space="preserve">r_:</s>
  <s xml:id="echoid-s12370" xml:space="preserve">: _f_. </s>
  <s xml:id="echoid-s12371" xml:space="preserve">{_rf_/_k_} = AK. </s>
  <s xml:id="echoid-s12372" xml:space="preserve">ergò {_rf_/_k_} - {_pma_/_rk_} = AL. </s>
  <s xml:id="echoid-s12373" xml:space="preserve">Et{_rrff_/_kk_} -<lb/>{2 _fmpa_/_kk_} (abjectis ſuperfluis) = AL_q_; </s>
  <s xml:id="echoid-s12374" xml:space="preserve">adeóque LF_q_ = <lb/>{_rrkk_ - _rrff_ + 2 _fmpa_/_kk_} = {_rrmm_ + 2 _fmpa_.</s>
  <s xml:id="echoid-s12375" xml:space="preserve">/_kk_}</s>
</p>
<div xml:id="echoid-div377" type="float" level="2" n="1">
<note position="right" xlink:label="note-0261-01" xlink:href="note-0261-01a" xml:space="preserve">Fig. 119.</note>
</div>
<p>
  <s xml:id="echoid-s12376" xml:space="preserve">Eſt autem AQ_q_. </s>
  <s xml:id="echoid-s12377" xml:space="preserve">QN_q_:</s>
  <s xml:id="echoid-s12378" xml:space="preserve">: AL_q_. </s>
  <s xml:id="echoid-s12379" xml:space="preserve">LF_q_; </s>
  <s xml:id="echoid-s12380" xml:space="preserve">hoc eſt Q: </s>
  <s xml:id="echoid-s12381" xml:space="preserve">_f_ - _e_. <lb/></s>
  <s xml:id="echoid-s12382" xml:space="preserve">Q: </s>
  <s xml:id="echoid-s12383" xml:space="preserve">_m_ + _a_:</s>
  <s xml:id="echoid-s12384" xml:space="preserve">: ALq. </s>
  <s xml:id="echoid-s12385" xml:space="preserve">LFq. </s>
  <s xml:id="echoid-s12386" xml:space="preserve">hoc eſt _ff_ - 2 _fe_. </s>
  <s xml:id="echoid-s12387" xml:space="preserve">_mm_ + 2 _ma_:</s>
  <s xml:id="echoid-s12388" xml:space="preserve">: <lb/>_rrff_ - 2 _fmpa_. </s>
  <s xml:id="echoid-s12389" xml:space="preserve">_rrmm_ + 2 _fmpa_. </s>
  <s xml:id="echoid-s12390" xml:space="preserve">Unde (ſublatis ex nor-<lb/>ma rejectaneis) emerget _æquatio_, _ffpa_ + _mmpa_ - _rrfa_ = _rrme_; </s>
  <s xml:id="echoid-s12391" xml:space="preserve">ſeu <lb/>_kkpa_ - _rrfa_ = _rrme_; </s>
  <s xml:id="echoid-s12392" xml:space="preserve">vel ſubſtituendo juxta _præſcriptum_; </s>
  <s xml:id="echoid-s12393" xml:space="preserve">_kkpm_ - _rrfm_ <lb/> = _rrmt_; </s>
  <s xml:id="echoid-s12394" xml:space="preserve">vel {_kkp_/_rr_} - _f_ = _t_. </s>
  <s xml:id="echoid-s12395" xml:space="preserve">Hinc colligitur eſſe rectam AT = <lb/>{_kk_/_rr_} _p_; </s>
  <s xml:id="echoid-s12396" xml:space="preserve">hoc eſt (quoniam, ut notum eſt, AV = {_rr_/_p_}) erit AT = <lb/>{AMq/AV}; </s>
  <s xml:id="echoid-s12397" xml:space="preserve">ſeu, AV. </s>
  <s xml:id="echoid-s12398" xml:space="preserve">AM:</s>
  <s xml:id="echoid-s12399" xml:space="preserve">: AM. </s>
  <s xml:id="echoid-s12400" xml:space="preserve">AT.</s>
  <s xml:id="echoid-s12401" xml:space="preserve"/>
</p>
<pb o="84" file="0262" n="277" rhead=""/>
</div>
<div xml:id="echoid-div379" type="section" level="1" n="40">
<head xml:id="echoid-head43" xml:space="preserve">Eæemp. V.</head>
<p>
  <s xml:id="echoid-s12402" xml:space="preserve">Sit DEB _Quadrans Circuli_, quem tangat recta BX; </s>
  <s xml:id="echoid-s12403" xml:space="preserve">tum linea <lb/>
<anchor type="note" xlink:label="note-0262-01a" xlink:href="note-0262-01"/>
AMO talis, ut in recta AV utcunque ſumptâ AP, quæ arcum BE <lb/>adæquet, erectáque PM ad AV normali, ſit PM æqualis arcûs BE <lb/>tangenti BG.</s>
  <s xml:id="echoid-s12404" xml:space="preserve"/>
</p>
<div xml:id="echoid-div379" type="float" level="2" n="1">
<note position="left" xlink:label="note-0262-01" xlink:href="note-0262-01a" xml:space="preserve">Fig. 120, <lb/>121.</note>
</div>
<p>
  <s xml:id="echoid-s12405" xml:space="preserve">Sumpto arcu BF = AQ: </s>
  <s xml:id="echoid-s12406" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12407" xml:space="preserve">ductâ CFH; </s>
  <s xml:id="echoid-s12408" xml:space="preserve">demiſſis EK, FL <lb/>ad CB normalibus; </s>
  <s xml:id="echoid-s12409" xml:space="preserve">nominentur CB = _r_. </s>
  <s xml:id="echoid-s12410" xml:space="preserve">CK = _f_: </s>
  <s xml:id="echoid-s12411" xml:space="preserve">KE = _g_. <lb/></s>
  <s xml:id="echoid-s12412" xml:space="preserve">Et quoniam eſt CE. </s>
  <s xml:id="echoid-s12413" xml:space="preserve">EK:</s>
  <s xml:id="echoid-s12414" xml:space="preserve">: arc. </s>
  <s xml:id="echoid-s12415" xml:space="preserve">EF. </s>
  <s xml:id="echoid-s12416" xml:space="preserve">LK; </s>
  <s xml:id="echoid-s12417" xml:space="preserve">vel CE. </s>
  <s xml:id="echoid-s12418" xml:space="preserve">EK:</s>
  <s xml:id="echoid-s12419" xml:space="preserve">: QF. </s>
  <s xml:id="echoid-s12420" xml:space="preserve"><lb/>LK; </s>
  <s xml:id="echoid-s12421" xml:space="preserve">hoc eſt _r_. </s>
  <s xml:id="echoid-s12422" xml:space="preserve">_g_:</s>
  <s xml:id="echoid-s12423" xml:space="preserve">: _e_. </s>
  <s xml:id="echoid-s12424" xml:space="preserve">{_ge_/_r_} = LK; </s>
  <s xml:id="echoid-s12425" xml:space="preserve">erit CL = _f_ + {_ge_.</s>
  <s xml:id="echoid-s12426" xml:space="preserve">/_r_} Et LF <lb/> = √ _rr_ - _ff_ - {2 _fge_/_r_} = √ _gg_ - {2 _fge_.</s>
  <s xml:id="echoid-s12427" xml:space="preserve">/_r_}</s>
</p>
<p>
  <s xml:id="echoid-s12428" xml:space="preserve">Eſt autem CL. </s>
  <s xml:id="echoid-s12429" xml:space="preserve">LF:</s>
  <s xml:id="echoid-s12430" xml:space="preserve">: (CB. </s>
  <s xml:id="echoid-s12431" xml:space="preserve">BH:</s>
  <s xml:id="echoid-s12432" xml:space="preserve">:) CB. </s>
  <s xml:id="echoid-s12433" xml:space="preserve">QN. </s>
  <s xml:id="echoid-s12434" xml:space="preserve">hoc eſt, <lb/>_f_ + {_ge_.</s>
  <s xml:id="echoid-s12435" xml:space="preserve">/_r_} √ _gg_ - {2 _fge_/_r_}:</s>
  <s xml:id="echoid-s12436" xml:space="preserve">: _r_. </s>
  <s xml:id="echoid-s12437" xml:space="preserve">_m_ - _a_. </s>
  <s xml:id="echoid-s12438" xml:space="preserve">vel (quadrando) _ff_ + <lb/>{2 _fge._</s>
  <s xml:id="echoid-s12439" xml:space="preserve">/_r_} _gg_ - {2 _fge_/_r_}:</s>
  <s xml:id="echoid-s12440" xml:space="preserve">: _rr._ </s>
  <s xml:id="echoid-s12441" xml:space="preserve">_mm_ - 2 _ma_. </s>
  <s xml:id="echoid-s12442" xml:space="preserve">Unde (dimiſſis quæ <lb/>oportet) obtinetur æquatio, _rfma_ = _grre_ + _gmme_. </s>
  <s xml:id="echoid-s12443" xml:space="preserve">unde <lb/>ſubſtituendo, eſt _rfmm_ = _grrt_ + _gmmt_. </s>
  <s xml:id="echoid-s12444" xml:space="preserve">vel {_rfmm_/_grr_ + _gmm_} = _t_. <lb/></s>
  <s xml:id="echoid-s12445" xml:space="preserve">ſeu (quoniam eſt _m_ = {_rg_/_f_}) erit _t_ = {_rr_/_rr_ + _mm_} _m_ = {CB_q_/CG_q_} BG = <lb/>{CK_q_/CE_q_} BG.</s>
  <s xml:id="echoid-s12446" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12447" xml:space="preserve">Hæc ſufficere videntur huic methodo elucidandæ.</s>
  <s xml:id="echoid-s12448" xml:space="preserve"/>
</p>
<pb o="85" file="0263" n="278"/>
</div>
<div xml:id="echoid-div381" type="section" level="1" n="41">
<head xml:id="echoid-head44" xml:space="preserve"><emph style="sc">Lect</emph>. XI.</head>
<p>
  <s xml:id="echoid-s12449" xml:space="preserve">R Eliquis utcunque patratis, apponemus iam _quæ ad magnitudinum_ <lb/>è _tangentibus_ (ſeu è perpendicularibus ad curvas) _Dimenſiones_ <lb/>_eliciendas pertinentia ſe objecerunt Tbeoremata_; </s>
  <s xml:id="echoid-s12450" xml:space="preserve">de compluribus utiq; <lb/></s>
  <s xml:id="echoid-s12451" xml:space="preserve">ſelectiora quædam.</s>
  <s xml:id="echoid-s12452" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12453" xml:space="preserve">I Sit curva quæpiam VH (cujus axis VD, applicata HD ad VD <lb/>normalis) item linea φZψ talis, ut ſi à curvæ puncto liberè ſumpto <lb/>
<anchor type="note" xlink:label="note-0263-01a" xlink:href="note-0263-01"/>
(putaE) ducatur recta EP ad curvam perpendicularis, &amp; </s>
  <s xml:id="echoid-s12454" xml:space="preserve">recta EAZ ad <lb/>axem perpenicularis, ſit recta AZ interceptæ AP æqualis; </s>
  <s xml:id="echoid-s12455" xml:space="preserve">erit _ſpatium_ <lb/>ADψφ_æq@ lis ſemiſſi quadr ati_ ex recta DH.</s>
  <s xml:id="echoid-s12456" xml:space="preserve"/>
</p>
<div xml:id="echoid-div381" type="float" level="2" n="1">
<note position="right" xlink:label="note-0263-01" xlink:href="note-0263-01a" xml:space="preserve">Fig. 122.</note>
</div>
<p>
  <s xml:id="echoid-s12457" xml:space="preserve">Nam ſit angulus HDO ſemirectus; </s>
  <s xml:id="echoid-s12458" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12459" xml:space="preserve">æquiſecetur recta V Din-<lb/>definitè punctis A, B, C; </s>
  <s xml:id="echoid-s12460" xml:space="preserve">per quæ ducantur rectæ EAZ, FBZ, <lb/>GCZ, ad HD parallelæ; </s>
  <s xml:id="echoid-s12461" xml:space="preserve">curvæ occurrentes in E, F, G; </s>
  <s xml:id="echoid-s12462" xml:space="preserve">à quibus <lb/>rectæ EIY, FKY, GLY ad VD (vel HO) parallelæ ducantur; <lb/></s>
  <s xml:id="echoid-s12463" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s12464" xml:space="preserve">rectæ EP, FP, GP, HP curvæ VH perpendiculares ſint; </s>
  <s xml:id="echoid-s12465" xml:space="preserve">li-<lb/>neæ verò ſe interſecent; </s>
  <s xml:id="echoid-s12466" xml:space="preserve">ut vides. </s>
  <s xml:id="echoid-s12467" xml:space="preserve">Eſtque triangulum HLG ſimile <lb/>triangulo PDH (nam ob indefinitam ſectionem curvula GH pro re-<lb/>ctà haberiporeſt) quare HL. </s>
  <s xml:id="echoid-s12468" xml:space="preserve">LG:</s>
  <s xml:id="echoid-s12469" xml:space="preserve">: PD. </s>
  <s xml:id="echoid-s12470" xml:space="preserve">DH. </s>
  <s xml:id="echoid-s12471" xml:space="preserve">adeóque HL x DH <lb/> = LG x PD; </s>
  <s xml:id="echoid-s12472" xml:space="preserve">hoc eſt HL x HO = DC x Dψ. </s>
  <s xml:id="echoid-s12473" xml:space="preserve">Simili monſtra <lb/>bitur diſcurſu, quoniam triangulum GMF triangulo PCG aſſimila-<lb/>tur, fore LK x LY = CB x CZ; </s>
  <s xml:id="echoid-s12474" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12475" xml:space="preserve">ſimiliter KI x KY = BA x <lb/>BZ; </s>
  <s xml:id="echoid-s12476" xml:space="preserve">itidem denuò ID x IY = AV x AZ; </s>
  <s xml:id="echoid-s12477" xml:space="preserve">unde conſtat triangu-<lb/>lum HDO (quod a rectangulis HL x HO + LK x LY + KI x <lb/>KY + ID x IY mi@mè differt) æqu@i ſoatio VDψφ (quod iti-<lb/>dem à rectangulis DC x Dψ + CB x CZ + BA x BZ + AV <lb/>x AZ minimè differt); </s>
  <s xml:id="echoid-s12478" xml:space="preserve">hoc eſt {DHq/2} æquari ſpatio VDψφ.</s>
  <s xml:id="echoid-s12479" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12480" xml:space="preserve">Longiordiſcurſus apagogicus adhiberi poſſit, at quorſum?</s>
  <s xml:id="echoid-s12481" xml:space="preserve"/>
</p>
<pb o="86" file="0264" n="279" rhead=""/>
<p>
  <s xml:id="echoid-s12482" xml:space="preserve">II. </s>
  <s xml:id="echoid-s12483" xml:space="preserve">Iiſdem poſitis, atque paratis; </s>
  <s xml:id="echoid-s12484" xml:space="preserve">_ſummarectangulorum_ AZ x AE <lb/>+ BZ x BF + CZ x CG, &amp;</s>
  <s xml:id="echoid-s12485" xml:space="preserve">e. </s>
  <s xml:id="echoid-s12486" xml:space="preserve">æquatur _trienti cubi_ ex baſe <lb/>
<anchor type="note" xlink:label="note-0264-01a" xlink:href="note-0264-01"/>
DH.</s>
  <s xml:id="echoid-s12487" xml:space="preserve"/>
</p>
<div xml:id="echoid-div382" type="float" level="2" n="2">
<note position="left" xlink:label="note-0264-01" xlink:href="note-0264-01a" xml:space="preserve">Fig. 122.</note>
</div>
<p>
  <s xml:id="echoid-s12488" xml:space="preserve">Nam ob HL. </s>
  <s xml:id="echoid-s12489" xml:space="preserve">LG:</s>
  <s xml:id="echoid-s12490" xml:space="preserve">: PD. </s>
  <s xml:id="echoid-s12491" xml:space="preserve">DH:</s>
  <s xml:id="echoid-s12492" xml:space="preserve">: PD x DH. </s>
  <s xml:id="echoid-s12493" xml:space="preserve">DHq; </s>
  <s xml:id="echoid-s12494" xml:space="preserve">erit HL x <lb/>DHq = LG x PD x DH. </s>
  <s xml:id="echoid-s12495" xml:space="preserve">hoc eſt HL x HOq = DC x Dψ x <lb/>DH. </s>
  <s xml:id="echoid-s12496" xml:space="preserve">Similíque diſcurſu, LK x LYq = CB x CZ x CG. </s>
  <s xml:id="echoid-s12497" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12498" xml:space="preserve">KI <lb/>x KYq = BA x BZ x BF, &amp;</s>
  <s xml:id="echoid-s12499" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12500" xml:space="preserve">Verùm HL x HOq + LK x <lb/>LYq + KI x KYq, &amp;</s>
  <s xml:id="echoid-s12501" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12502" xml:space="preserve">adæquant trientem cubi ex DH; </s>
  <s xml:id="echoid-s12503" xml:space="preserve">itaque <lb/>liquet Propoſitum.</s>
  <s xml:id="echoid-s12504" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12505" xml:space="preserve">III. </s>
  <s xml:id="echoid-s12506" xml:space="preserve">Simili ratione conſtabit ſummam AZ x AEq + BZ x BFq <lb/>+ CZ x CGq, &amp;</s>
  <s xml:id="echoid-s12507" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12508" xml:space="preserve">æquari τῶ{DH_qq_;</s>
  <s xml:id="echoid-s12509" xml:space="preserve">/4} &amp; </s>
  <s xml:id="echoid-s12510" xml:space="preserve">eſſe ſummam AZ x <lb/>AE cub. </s>
  <s xml:id="echoid-s12511" xml:space="preserve">+ BZ x BE cub. </s>
  <s xml:id="echoid-s12512" xml:space="preserve">+ CZ x CG cub &amp;</s>
  <s xml:id="echoid-s12513" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12514" xml:space="preserve">= {DH {5/ }/5}; </s>
  <s xml:id="echoid-s12515" xml:space="preserve">ac <lb/>eodem in continuum tenore.</s>
  <s xml:id="echoid-s12516" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12517" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s12518" xml:space="preserve">Exhinc conſectantur haud aſpernanda _Theoremata_: </s>
  <s xml:id="echoid-s12519" xml:space="preserve">Sit <lb/>VDψφ ſpatium quodlibet, cujus axis VD, ut dictum, æquiſectus; <lb/></s>
  <s xml:id="echoid-s12520" xml:space="preserve">
<anchor type="note" xlink:label="note-0264-02a" xlink:href="note-0264-02"/>
ſi concipiantur ſingula ſpatia VAZφ, VBZφ, VCZφ, &amp;</s>
  <s xml:id="echoid-s12521" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12522" xml:space="preserve">in <lb/>ſuas ordinatas AZ, BZ, CZ, &amp;</s>
  <s xml:id="echoid-s12523" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12524" xml:space="preserve">reſpectivè ſingulas duci, quæ pro-<lb/>veniet ſumma adæquabitur ipſius ſpatii VDψφ ſemiquadrato.</s>
  <s xml:id="echoid-s12525" xml:space="preserve"/>
</p>
<div xml:id="echoid-div383" type="float" level="2" n="3">
<note position="left" xlink:label="note-0264-02" xlink:href="note-0264-02a" xml:space="preserve">Fig. 122.</note>
</div>
<p>
  <s xml:id="echoid-s12526" xml:space="preserve">Nam (utì priùs oſtenſum) figuræ VDψ φ adaptari poteſt ſpatium <lb/>VDH; </s>
  <s xml:id="echoid-s12527" xml:space="preserve">tale nimirum ut ductà quâvis ad curvam VH perpendiculari, <lb/>ceu EP, ſit AP ſibireſpondenti applicatæ AZ æqualis; </s>
  <s xml:id="echoid-s12528" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_b_)"/> unde fiet
<anchor type="note" xlink:label="note-0264-03a" xlink:href="note-0264-03"/>
<anchor type="note" xlink:label="note-0264-04a" xlink:href="note-0264-04"/>
<anchor type="note" xlink:label="note-0264-05a" xlink:href="note-0264-05"/>
ſpatium VAZ φ = {AE_q_/2}; </s>
  <s xml:id="echoid-s12529" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12530" xml:space="preserve">VBZ φ = {BF_q_;</s>
  <s xml:id="echoid-s12531" xml:space="preserve">/2} &amp; </s>
  <s xml:id="echoid-s12532" xml:space="preserve">VCZ φ = {CG_q_/2} <lb/>&amp;</s>
  <s xml:id="echoid-s12533" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12534" xml:space="preserve">quapropter omnia VAZφ x AZ + VBZφ x BZ + VCZφ <lb/>x CZ, &amp;</s>
  <s xml:id="echoid-s12535" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12536" xml:space="preserve">æquabuntur omnibus {AE_q_ x AZ + BF_q_ x BZ + CG_q_ x CZ/2} <lb/>
<anchor type="note" xlink:href="" symbol="(_c_)"/> hoc eſt τῶ {DH_qq_/4 x 2}; </s>
  <s xml:id="echoid-s12537" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_b_)"/> hoc eſt τῶ {VDψφ x VDψφ/2.</s>
  <s xml:id="echoid-s12538" xml:space="preserve">}</s>
</p>
<div xml:id="echoid-div384" type="float" level="2" n="4">
<note position="left" xlink:label="note-0264-03" xlink:href="note-0264-03a" xml:space="preserve">_Præced_. Lect. X.</note>
<note symbol="(_b_)" position="left" xlink:label="note-0264-04" xlink:href="note-0264-04a" xml:space="preserve">1 _hujus_.</note>
<note position="left" xlink:label="note-0264-05" xlink:href="note-0264-05a" xml:space="preserve">Fig. 122.</note>
</div>
<note symbol="(_c_)" position="left" xml:space="preserve">3 _hujus_.</note>
<p>
  <s xml:id="echoid-s12539" xml:space="preserve">V. </s>
  <s xml:id="echoid-s12540" xml:space="preserve">Quòd ſi ducantur omnia √ VAZφ, √ VBZφ, √ VCZφ, <lb/>&amp;</s>
  <s xml:id="echoid-s12541" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12542" xml:space="preserve">in ſuas applicatas AZ, BZ, CZ, &amp;</s>
  <s xml:id="echoid-s12543" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12544" xml:space="preserve">reſpectivè proveniet ag-<lb/>gregatum æquale duabus tertiis radicis quadratæ facti ex ipſo ſpatio <lb/>VDψφ cubato (τῶ {2/3} √ VDψφ {3/ })</s>
</p>
<p>
  <s xml:id="echoid-s12545" xml:space="preserve">Nam adaptatâ curvâ VH, eſt √ VAZ φ = AE√ {1/2}; </s>
  <s xml:id="echoid-s12546" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12547" xml:space="preserve">√ VBZφ <lb/> = BF √ {1/2}, &amp; </s>
  <s xml:id="echoid-s12548" xml:space="preserve">VCZφ = √ CG √ {1/2}, &amp;</s>
  <s xml:id="echoid-s12549" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12550" xml:space="preserve">Cùm itaque ſint
<pb o="87" file="0265" n="280" rhead=""/>
omnia AZ x AE + BZ x BF + CZ x CG, &amp;</s>
  <s xml:id="echoid-s12551" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12552" xml:space="preserve">= {DHcub.</s>
  <s xml:id="echoid-s12553" xml:space="preserve">/3} <lb/>crunt omnia AZ x √ VAZ φ + BZ x √ VBZφ + CZ x <lb/>√ VCZφ, &amp;</s>
  <s xml:id="echoid-s12554" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12555" xml:space="preserve">= {DHcub/3} √{1/2} = √ {DH<emph style="sub">6</emph>/18.</s>
  <s xml:id="echoid-s12556" xml:space="preserve">} Eſt autem DH_q_ = <lb/>2 VD ψ φ, vel DH<emph style="sub">6</emph> = 8VD ψ φ {3/ }; </s>
  <s xml:id="echoid-s12557" xml:space="preserve">quapropter omnia AZ x <lb/>√ VAZ φ + BZ x √ VBZφ + CZ x √ VCZφ, &amp;</s>
  <s xml:id="echoid-s12558" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12559" xml:space="preserve">= √ <lb/>{8/18} VD ψ φ {3/ } = {2/3} √ VDψφ {3/ }.</s>
  <s xml:id="echoid-s12560" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12561" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s12562" xml:space="preserve">_Exempla._ </s>
  <s xml:id="echoid-s12563" xml:space="preserve">Sit VDψ circuli quadrans (cujus radius dicatur <lb/>R, &amp; </s>
  <s xml:id="echoid-s12564" xml:space="preserve">Peripheria P) ſegmenta VAZ, VBZ, VCZ, &amp;</s>
  <s xml:id="echoid-s12565" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12566" xml:space="preserve">in ſi-<lb/>nus rectos AZ, BZ, CZ, &amp;</s>
  <s xml:id="echoid-s12567" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12568" xml:space="preserve">ducta conficient {R_q_P_q_/8.</s>
  <s xml:id="echoid-s12569" xml:space="preserve">}</s>
</p>
<note position="right" xml:space="preserve">Fig. 123.</note>
<p>
  <s xml:id="echoid-s12570" xml:space="preserve">Item Summa AZ √ VAZ + BZ √ VBZ + CZ √ VCZ, <lb/>&amp;</s>
  <s xml:id="echoid-s12571" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12572" xml:space="preserve">= {2/3} √{R<emph style="sub">3</emph>P<emph style="sub">3</emph>/8.</s>
  <s xml:id="echoid-s12573" xml:space="preserve">} = √ {R<emph style="sub">3</emph>P<emph style="sub">3</emph>/18.</s>
  <s xml:id="echoid-s12574" xml:space="preserve">}</s>
</p>
<p>
  <s xml:id="echoid-s12575" xml:space="preserve">Si VD ψ ſit parabolæ ſegmentum, factum è ſegmentis in applicatas <lb/>erit {2/9} VD_q_ x Aψ_q_; </s>
  <s xml:id="echoid-s12576" xml:space="preserve">ac è radicibus ſegmentorum in applicatas factum <lb/>erit {2/3} √ {8/2} { /7} VD<emph style="sub">3</emph> x Dψ<emph style="sub">3</emph> √ {3/2} {2/4} { /3} VD<emph style="sub">3</emph> x Dψ<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s12577" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12578" xml:space="preserve">Similia plura de factis è _Segmentorum poteſtatibus, autradicibus_ <lb/>_aliis in applicatas, aut ſinus ductis_, hinc extundi poſſent.</s>
  <s xml:id="echoid-s12579" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12580" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s12581" xml:space="preserve">E dictis porrò ſequitur, ſi omnes ( vertici, &amp; </s>
  <s xml:id="echoid-s12582" xml:space="preserve">perpendicula-<lb/>ribus interjectæ) VP per reſpectiva puncta A, B, C, &amp;</s>
  <s xml:id="echoid-s12583" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12584" xml:space="preserve">Concipian-<lb/>tur applicatæ, puta ut AY, BY, CY, &amp;</s>
  <s xml:id="echoid-s12585" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12586" xml:space="preserve">reſpectivis VP æquentur; <lb/></s>
  <s xml:id="echoid-s12587" xml:space="preserve">erit è ſic applicatis _conſtitutum ſpatium_ ADξθ _æquale ſemiſſe quadrati_ <lb/>_ex ſubtenſa_ VH.</s>
  <s xml:id="echoid-s12588" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12589" xml:space="preserve">Nam, ob omnes VA + VB + VC, &amp;</s>
  <s xml:id="echoid-s12590" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12591" xml:space="preserve">= {VD_q_;</s>
  <s xml:id="echoid-s12592" xml:space="preserve">/2} &amp; </s>
  <s xml:id="echoid-s12593" xml:space="preserve">omnes <lb/>AP + BP + CP&amp;</s>
  <s xml:id="echoid-s12594" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12595" xml:space="preserve">= {DH_q_,/2} liquet fore omnes VP = <lb/>{VH_q_.</s>
  <s xml:id="echoid-s12596" xml:space="preserve">/2}</s>
</p>
<p>
  <s xml:id="echoid-s12597" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s12598" xml:space="preserve">Porrò, ſi (poſitis iiſdem) ſit curva RXXS talis, ut ſit IX <lb/> = AP, &amp; </s>
  <s xml:id="echoid-s12599" xml:space="preserve">KX = BP; </s>
  <s xml:id="echoid-s12600" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12601" xml:space="preserve">LX = CP, &amp;</s>
  <s xml:id="echoid-s12602" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12603" xml:space="preserve">erit _ſolidum factum ex_
<pb o="88" file="0266" n="281" rhead=""/>
_ſpatio_ VDψ φ _circa axem_ VD _rotato ſubduplnm ſolidi ex ſpatio_ <lb/>DRSH, _itidem circa axem_ VD _rotato, confecti._</s>
  <s xml:id="echoid-s12604" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12605" xml:space="preserve">Nam ob HL. </s>
  <s xml:id="echoid-s12606" xml:space="preserve">LG:</s>
  <s xml:id="echoid-s12607" xml:space="preserve">: PD. </s>
  <s xml:id="echoid-s12608" xml:space="preserve">DH:</s>
  <s xml:id="echoid-s12609" xml:space="preserve">: Dψ. </s>
  <s xml:id="echoid-s12610" xml:space="preserve">DH:</s>
  <s xml:id="echoid-s12611" xml:space="preserve">: Dψ_q_. </s>
  <s xml:id="echoid-s12612" xml:space="preserve">Dψ x <lb/>DH:</s>
  <s xml:id="echoid-s12613" xml:space="preserve">: Dψ_q_. </s>
  <s xml:id="echoid-s12614" xml:space="preserve">HS x DH; </s>
  <s xml:id="echoid-s12615" xml:space="preserve">erit HL x HS x DH = LG x Dψ_q_. <lb/></s>
  <s xml:id="echoid-s12616" xml:space="preserve"> = DC x Dψ_q_. </s>
  <s xml:id="echoid-s12617" xml:space="preserve">Simili planè diſcurſu erit LK x LX x DL = <lb/>CB x CZ_q_; </s>
  <s xml:id="echoid-s12618" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12619" xml:space="preserve">KI x KX x DK = BA x BZ_q_, &amp;</s>
  <s xml:id="echoid-s12620" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12621" xml:space="preserve">atqui ſoli-<lb/>dum pt<unsure/>ius eſt {ῶ/δ}: </s>
  <s xml:id="echoid-s12622" xml:space="preserve">AZ_q_ + BZ_q_ + CZ_q_, &amp;</s>
  <s xml:id="echoid-s12623" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12624" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12625" xml:space="preserve">ſolidum poſte-<lb/>rius eſt {2 ῶ/δ}: </s>
  <s xml:id="echoid-s12626" xml:space="preserve">DI x IX + DK x KX + DL x LX, &amp;</s>
  <s xml:id="echoid-s12627" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12628" xml:space="preserve">itaque <lb/>conſtat Propoſitum.</s>
  <s xml:id="echoid-s12629" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12630" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s12631" xml:space="preserve">Hæc itidem omnia ſimili ratione vera ſunt, etiam ſi curva VEH <lb/>
<anchor type="note" xlink:label="note-0266-01a" xlink:href="note-0266-01"/>
rectæ VD convexas ſuas partes obvertat; </s>
  <s xml:id="echoid-s12632" xml:space="preserve">nempe quovis in curva ac-<lb/>cepto puncto E; </s>
  <s xml:id="echoid-s12633" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12634" xml:space="preserve">per hoc ductâ EP ad curvam VEH perpendicu-<lb/>lari, &amp; </s>
  <s xml:id="echoid-s12635" xml:space="preserve">EAY ad rectam VD normali, factáque AZ = AP; </s>
  <s xml:id="echoid-s12636" xml:space="preserve">erit <lb/>ſpatium VDψ = {DH_q_;</s>
  <s xml:id="echoid-s12637" xml:space="preserve">/2} Sin quoque fiat AY = VP; </s>
  <s xml:id="echoid-s12638" xml:space="preserve">erit ſpati-<lb/>um VD ψ = {VH_q_;</s>
  <s xml:id="echoid-s12639" xml:space="preserve">/2} Et pariter quoad cætera.</s>
  <s xml:id="echoid-s12640" xml:space="preserve"/>
</p>
<div xml:id="echoid-div385" type="float" level="2" n="5">
<note position="left" xlink:label="note-0266-01" xlink:href="note-0266-01a" xml:space="preserve">Fig. 124.</note>
</div>
<p>
  <s xml:id="echoid-s12641" xml:space="preserve">Ex his verò _Theorematis quam innumerarum magnitudinum_ (ex <lb/>ipſarum immediatè conſtructione) _dimenſiones innoteſcant_, ab expe-<lb/>rientia facilè comperietur.</s>
  <s xml:id="echoid-s12642" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12643" xml:space="preserve">X. </s>
  <s xml:id="echoid-s12644" xml:space="preserve">Sit rurſus curva quæpiam VH (cujus axis VD, baſis DH) <lb/>
<anchor type="note" xlink:label="note-0266-02a" xlink:href="note-0266-02"/>
&amp; </s>
  <s xml:id="echoid-s12645" xml:space="preserve">linea DZZO talis, ut a curvæ puncto quopiam, cen E, ductâ <lb/>rectâ ET, quæ curvam tangat, &amp; </s>
  <s xml:id="echoid-s12646" xml:space="preserve">recta EIZ ad baſin parallelâ, ſit <lb/>qerpetuò IZ æqualis ipſi AT; </s>
  <s xml:id="echoid-s12647" xml:space="preserve">dico _ſpatium_ DHO _ſpatio_ VDH <lb/>_æquari_.</s>
  <s xml:id="echoid-s12648" xml:space="preserve"/>
</p>
<div xml:id="echoid-div386" type="float" level="2" n="6">
<note position="left" xlink:label="note-0266-02" xlink:href="note-0266-02a" xml:space="preserve">Fig. 125.</note>
</div>
<p>
  <s xml:id="echoid-s12649" xml:space="preserve">Æquiſecetur enim recta DH indefinitè, punctis I, K, L, per quæ <lb/>ducantur rectæ EIZ, FKZ, GLZ ad VD parallelæ, curvæque oc-<lb/>currentes ad E, F, G, unde ducantur rectæ EA, FB, GC ad HD <lb/>parallelæ, rectæque ET, FT, GT (ut &amp; </s>
  <s xml:id="echoid-s12650" xml:space="preserve">HT) _curvam tangentes;_ <lb/></s>
  <s xml:id="echoid-s12651" xml:space="preserve">lineæ verò ſe, ut Schema monſtrat, interſecent. </s>
  <s xml:id="echoid-s12652" xml:space="preserve">Eſtque jam triangu-<lb/>lum GLH ſimile triangulo TDH (nam ob diviſionem iſtam indefi-<lb/>nitam arculus GH rectæ inſtar cenſeri poteſt, eatenus tangenti HT <lb/>coincidens) quare LG. </s>
  <s xml:id="echoid-s12653" xml:space="preserve">LH:</s>
  <s xml:id="echoid-s12654" xml:space="preserve">: TD. </s>
  <s xml:id="echoid-s12655" xml:space="preserve">DH; </s>
  <s xml:id="echoid-s12656" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12657" xml:space="preserve">LG x DH = LH <lb/>x TD; </s>
  <s xml:id="echoid-s12658" xml:space="preserve">ſeu CD x DH = LH x HO. </s>
  <s xml:id="echoid-s12659" xml:space="preserve">ſimili ratiocinio eſt BC x
<pb o="89" file="0267" n="282" rhead=""/>
CG = KL x LZ; </s>
  <s xml:id="echoid-s12660" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12661" xml:space="preserve">AB x BF = IK x KZ, &amp; </s>
  <s xml:id="echoid-s12662" xml:space="preserve">VA x AE = <lb/>DI x IZ. </s>
  <s xml:id="echoid-s12663" xml:space="preserve">Verùm ſumma CD x DH + BC x CG + AB x <lb/>BF + VA x AE à ſpatio VDH minimè differt; </s>
  <s xml:id="echoid-s12664" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12665" xml:space="preserve">ſumma LH x <lb/>DO + KL x LZ + IK x KZ + DI x IZ à ſpatio DHO mi-<lb/>nimè differt. </s>
  <s xml:id="echoid-s12666" xml:space="preserve">itaque ſpatio VDH, DHO æquantur.</s>
  <s xml:id="echoid-s12667" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12668" xml:space="preserve">Hoc _perutile Theorema_ doctiſſimo Viro D. </s>
  <s xml:id="echoid-s12669" xml:space="preserve">_Gregorio Aberdonenſi_ <lb/>debetur; </s>
  <s xml:id="echoid-s12670" xml:space="preserve">cui ſequentia ſubnectimus.</s>
  <s xml:id="echoid-s12671" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12672" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s12673" xml:space="preserve">Iiſdem poſitis; </s>
  <s xml:id="echoid-s12674" xml:space="preserve">ſolidum ex ſpatio DHO circa axem VDR <lb/>rotato factum duplum erit ſolidi facti ex ſpatio VDH itidem circa ax-<lb/>
<anchor type="note" xlink:label="note-0267-01a" xlink:href="note-0267-01"/>
em VD rotato.</s>
  <s xml:id="echoid-s12675" xml:space="preserve"/>
</p>
<div xml:id="echoid-div387" type="float" level="2" n="7">
<note position="right" xlink:label="note-0267-01" xlink:href="note-0267-01a" xml:space="preserve">Fig. 125.</note>
</div>
<p>
  <s xml:id="echoid-s12676" xml:space="preserve">Nam eſt HL. </s>
  <s xml:id="echoid-s12677" xml:space="preserve">LG:</s>
  <s xml:id="echoid-s12678" xml:space="preserve">: (DH. </s>
  <s xml:id="echoid-s12679" xml:space="preserve">DT:</s>
  <s xml:id="echoid-s12680" xml:space="preserve">: DH. </s>
  <s xml:id="echoid-s12681" xml:space="preserve">HO:</s>
  <s xml:id="echoid-s12682" xml:space="preserve">:) DHq. <lb/></s>
  <s xml:id="echoid-s12683" xml:space="preserve">DH x HO. </s>
  <s xml:id="echoid-s12684" xml:space="preserve">unde HL x DH x HO = LG x DHq = CD x <lb/>DHq. </s>
  <s xml:id="echoid-s12685" xml:space="preserve">Similíque diſcurſu ſunt LK x DL x LZ = BC x CGq. </s>
  <s xml:id="echoid-s12686" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s12687" xml:space="preserve">KI x DK x KZ = AB x BFq. </s>
  <s xml:id="echoid-s12688" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12689" xml:space="preserve">demum ID x DI x IZ = <lb/>VA x AEq. </s>
  <s xml:id="echoid-s12690" xml:space="preserve">Eſt autem (ut vulgò notatum habetur) ſumma CD <lb/>x DHq + BCB x CGq + AB x BFq + VA x AEq dupla <lb/>ſummæ DI x IE + DK x KF + DL x LG, &amp;</s>
  <s xml:id="echoid-s12691" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12692" xml:space="preserve">Quare ſolidum <lb/>ex ſpatio HDO circa axem DR converſo factum duplum eſt ſolidi, <lb/>quod è ſpatio VDH circa VD converſo producitur.</s>
  <s xml:id="echoid-s12693" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12694" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s12695" xml:space="preserve">Hinc, ſumma DI x IZ + DK x KZ + DL x LZ, &amp;</s>
  <s xml:id="echoid-s12696" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s12697" xml:space="preserve">æquatur ſummæ quadratorum ex applicatis ad VD; </s>
  <s xml:id="echoid-s12698" xml:space="preserve">ſcilicet ipſis AEq <lb/>+ BFq + CGq, &amp;</s>
  <s xml:id="echoid-s12699" xml:space="preserve">c.</s>
  <s xml:id="echoid-s12700" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12701" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s12702" xml:space="preserve">Simili ratiocinio conſtabit ſummam DIq x IZ + DKq x <lb/>KZ + DLq x LZ, &amp;</s>
  <s xml:id="echoid-s12703" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12704" xml:space="preserve">triplam eſſe ſummæ DIq x IE + DKq <lb/>x KF + DLq x LG, &amp;</s>
  <s xml:id="echoid-s12705" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12706" xml:space="preserve">hòc eſt æqualem ſummæ cuborum ab <lb/>omnibus AE, BF, CG, &amp;</s>
  <s xml:id="echoid-s12707" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12708" xml:space="preserve">ad VD applicatis. </s>
  <s xml:id="echoid-s12709" xml:space="preserve">Idem quoad _re-_ <lb/>_liquas poteſtates_ obſervabilis eſt Concluſionum tenor.</s>
  <s xml:id="echoid-s12710" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12711" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s12712" xml:space="preserve">Iiſdem poſitis; </s>
  <s xml:id="echoid-s12713" xml:space="preserve">ſi DXH ſit linea talis, ut quævis ad DH <lb/>o<unsure/>rdinata, ceu IX, ſit media proportionalis inter ſibi congruas ordi-<lb/>natas IE, IZ; </s>
  <s xml:id="echoid-s12714" xml:space="preserve">erìt ſolidum ex ſpatio VDH circa axem DH rotato <lb/>duplum ſolidi ex ſpatio DXH circa eundem axem DH converſo pro-<lb/>creati.</s>
  <s xml:id="echoid-s12715" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12716" xml:space="preserve">Nam ob VA x AE = DI x IZ, erit VA x AE x EI = DI x IZ x IE = ID x <lb/>IXq. </s>
  <s xml:id="echoid-s12717" xml:space="preserve">Similíque de cauſa AB x BF x FK = IK x KXq; </s>
  <s xml:id="echoid-s12718" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12719" xml:space="preserve">BC <lb/>
<anchor type="note" xlink:label="note-0267-02a" xlink:href="note-0267-02"/>
x CG x GL = KL x LXq, &amp;</s>
  <s xml:id="echoid-s12720" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12721" xml:space="preserve">Eſt autem ſumma VA x AE <lb/>x EI + AB x BF x FK + BC x CG x GL, &amp;</s>
  <s xml:id="echoid-s12722" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12723" xml:space="preserve">Subdupla ſum-
<pb o="90" file="0268" n="283" rhead=""/>
mæ VDq + EIq + FKq + GLq; </s>
  <s xml:id="echoid-s12724" xml:space="preserve">ergò ſumma IXq + <lb/>KXq + LXq + HXq, ſubdupla eſt ſummæ VDq - EIq + <lb/>FKq + GLq. </s>
  <s xml:id="echoid-s12725" xml:space="preserve">Vnde liquet Propoſitum.</s>
  <s xml:id="echoid-s12726" xml:space="preserve"/>
</p>
<div xml:id="echoid-div388" type="float" level="2" n="8">
<note position="right" xlink:label="note-0267-02" xlink:href="note-0267-02a" xml:space="preserve">In 10. hujus.</note>
</div>
<p>
  <s xml:id="echoid-s12727" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s12728" xml:space="preserve">Quòd ſi curva DXH talis concipiatur, ut ſit ordinata quæpiam, <lb/>ceu IX, inter congruas ordinatas IE, IZ bimedia *; </s>
  <s xml:id="echoid-s12729" xml:space="preserve">erit ſumma cubo-<lb/>rum ex IX, KX, LX, &amp;</s>
  <s xml:id="echoid-s12730" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12731" xml:space="preserve">ſubtripla cuborum ex DV, IE, KF, &amp;</s>
  <s xml:id="echoid-s12732" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12733" xml:space="preserve">Sin IX <lb/>ſit trimed. </s>
  <s xml:id="echoid-s12734" xml:space="preserve">* erit IXqq + KXqq + LXqq, &amp;</s>
  <s xml:id="echoid-s12735" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12736" xml:space="preserve">= {DVqq + IEqq + KFqq/4} <lb/>&amp;</s>
  <s xml:id="echoid-s12737" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12738" xml:space="preserve">ac ità porrò quoad cæteras poteſtates. </s>
  <s xml:id="echoid-s12739" xml:space="preserve">* _Not._ </s>
  <s xml:id="echoid-s12740" xml:space="preserve">bimediam ap-<lb/>pello, quæ duarum mediarum proportionalium prima; </s>
  <s xml:id="echoid-s12741" xml:space="preserve">trimediam, <lb/>quæ trium prima eſt, &amp;</s>
  <s xml:id="echoid-s12742" xml:space="preserve">c.</s>
  <s xml:id="echoid-s12743" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12744" xml:space="preserve">Hæc ſimili ratione colliguntur, ac comprobantur. </s>
  <s xml:id="echoid-s12745" xml:space="preserve">piget χοχχὺζι<unsure/>ν.</s>
  <s xml:id="echoid-s12746" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12747" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s12748" xml:space="preserve">Sit porrò linea VYQ talis, ut ordinata AY ipſi AT; </s>
  <s xml:id="echoid-s12749" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12750" xml:space="preserve"><lb/>ordinata BY ipſi BT, &amp;</s>
  <s xml:id="echoid-s12751" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12752" xml:space="preserve">æquentur; </s>
  <s xml:id="echoid-s12753" xml:space="preserve">erit IZq + KZq + LZq, <lb/>&amp;</s>
  <s xml:id="echoid-s12754" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12755" xml:space="preserve">(ſumma quadratorum ex ordinatis à curva DZO ad rectam DH) <lb/>æqualis ſummæ VA x AE x AY + AB x BF x BY + BC x CG <lb/>x CY, &amp;</s>
  <s xml:id="echoid-s12756" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12757" xml:space="preserve">(hoc eſt figuræ VDH in figuram VDQ ductæ).</s>
  <s xml:id="echoid-s12758" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12759" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s12760" xml:space="preserve">Item, ſumma IZ. </s>
  <s xml:id="echoid-s12761" xml:space="preserve">cub. </s>
  <s xml:id="echoid-s12762" xml:space="preserve">+ KZ cub. </s>
  <s xml:id="echoid-s12763" xml:space="preserve">+ LZ cub. </s>
  <s xml:id="echoid-s12764" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s12765" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12766" xml:space="preserve">= <lb/>VA x AE x AYq + AB x BE x BYq + BC x CG x CYq, <lb/>&amp;</s>
  <s xml:id="echoid-s12767" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12768" xml:space="preserve">_hoc eſt figuræ_ VDH _in figuræ_ VDQ _quadrata ductæ_). </s>
  <s xml:id="echoid-s12769" xml:space="preserve">Simi-<lb/>lis &amp; </s>
  <s xml:id="echoid-s12770" xml:space="preserve">aliarum _poteſtatum_ eſt ratio.</s>
  <s xml:id="echoid-s12771" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12772" xml:space="preserve">Ad ſuperiorum normam hæc facilè colliges.</s>
  <s xml:id="echoid-s12773" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12774" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s12775" xml:space="preserve">Eadem vera ſunt, &amp; </s>
  <s xml:id="echoid-s12776" xml:space="preserve">omnino ſimiliratione comprobantur, <lb/>Etiam ſi curvæ VH convexa rectæ VD obvertantur. </s>
  <s xml:id="echoid-s12777" xml:space="preserve">Nempe, ſi linea <lb/>
<anchor type="note" xlink:label="note-0268-01a" xlink:href="note-0268-01"/>
DZO talis ſit, ut ductâ per quodvis in curva VH punctum E tangente <lb/>ET, &amp; </s>
  <s xml:id="echoid-s12778" xml:space="preserve">EA ad HD parallelâ, ac EIZ ad VD parallelâ, ſit perpetim IZ = <lb/>AT; </s>
  <s xml:id="echoid-s12779" xml:space="preserve">erit ſpatium DHO ſpatio VDH æquale; </s>
  <s xml:id="echoid-s12780" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12781" xml:space="preserve">ſolidum factum ex ſpa-<lb/>DHO circa axem VR converſo duplum erit ſolidi ex ſpatio VDH <lb/>circa eundem axem VD rotato producti. </s>
  <s xml:id="echoid-s12782" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s12783" xml:space="preserve">reliqua pari modo <lb/>convenient.</s>
  <s xml:id="echoid-s12784" xml:space="preserve"/>
</p>
<div xml:id="echoid-div389" type="float" level="2" n="9">
<note position="left" xlink:label="note-0268-01" xlink:href="note-0268-01a" xml:space="preserve">Fig. 126.</note>
</div>
<p>
  <s xml:id="echoid-s12785" xml:space="preserve">XIX Porrò, ſit curva quæpiam AMB, cujus axis AD, &amp; </s>
  <s xml:id="echoid-s12786" xml:space="preserve">huic <lb/>perpendicularis BD; </s>
  <s xml:id="echoid-s12787" xml:space="preserve">tum alia ſit linea KZL talis, ut ſumpto in cur-<lb/>va AB utcunque puncto M; </s>
  <s xml:id="echoid-s12788" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12789" xml:space="preserve">per hoc ductis rectâ MT curvam <lb/>
<anchor type="note" xlink:label="note-0268-02a" xlink:href="note-0268-02"/>
AB tangente, rectâ MFZ ad DB parallelâ (quæ lineam KL ſecet <lb/>in Z, rectam AD in F) datâque quâdam lineâ R; </s>
  <s xml:id="echoid-s12790" xml:space="preserve">ſit TF. </s>
  <s xml:id="echoid-s12791" xml:space="preserve">FM:</s>
  <s xml:id="echoid-s12792" xml:space="preserve">:
<pb o="91" file="0269" n="284" rhead=""/>
R. </s>
  <s xml:id="echoid-s12793" xml:space="preserve">FZ; </s>
  <s xml:id="echoid-s12794" xml:space="preserve">erit ſpatium ADLK æquale rectangulo ex R, &amp; </s>
  <s xml:id="echoid-s12795" xml:space="preserve">DB.</s>
  <s xml:id="echoid-s12796" xml:space="preserve"/>
</p>
<div xml:id="echoid-div390" type="float" level="2" n="10">
<note position="left" xlink:label="note-0268-02" xlink:href="note-0268-02a" xml:space="preserve">Fig. 127.</note>
</div>
<p>
  <s xml:id="echoid-s12797" xml:space="preserve">Nam ſit DH = R; </s>
  <s xml:id="echoid-s12798" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12799" xml:space="preserve">compleatur rectangulum BDHI; </s>
  <s xml:id="echoid-s12800" xml:space="preserve">tum <lb/>aſſumptâ MN indeſinitè parvâ curvæ AB partìculâ ducantur NG ad <lb/>BD; </s>
  <s xml:id="echoid-s12801" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12802" xml:space="preserve">MEX, NOS ad AD parallelæ. </s>
  <s xml:id="echoid-s12803" xml:space="preserve">Eſtque NO. </s>
  <s xml:id="echoid-s12804" xml:space="preserve">MO:</s>
  <s xml:id="echoid-s12805" xml:space="preserve">: <lb/>TF. </s>
  <s xml:id="echoid-s12806" xml:space="preserve">FM:</s>
  <s xml:id="echoid-s12807" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s12808" xml:space="preserve">FZ. </s>
  <s xml:id="echoid-s12809" xml:space="preserve">Unde NO x FZ = MO x R; </s>
  <s xml:id="echoid-s12810" xml:space="preserve">hoc eſt FG <lb/>x FZ = ES x EX. </s>
  <s xml:id="echoid-s12811" xml:space="preserve">ergò cum omnia rectangula FG x FZ minimè <lb/>differant à ſpatio ADLK; </s>
  <s xml:id="echoid-s12812" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12813" xml:space="preserve">omnia totidem rectangula ES x EX <lb/>componant rectangulum DHIB, ſatìs liquet Propoſitum.</s>
  <s xml:id="echoid-s12814" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12815" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s12816" xml:space="preserve">Iiſdem poſitis, ſit curva PYQ talis, ut ſumpta in ſumpta <lb/>recta MX ordinata EY (reſpectivæ) ipſi FZ æquetur, erit _ſumma_ <lb/>_quadr atorum_ ex FZ (ad rectam AD computata) par ei quod fit ex <lb/>ipſa R in _ſpatium_ DBQB ducta.</s>
  <s xml:id="echoid-s12817" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12818" xml:space="preserve">Eſt enim FG. </s>
  <s xml:id="echoid-s12819" xml:space="preserve">ES:</s>
  <s xml:id="echoid-s12820" xml:space="preserve">: NO. </s>
  <s xml:id="echoid-s12821" xml:space="preserve">MO:</s>
  <s xml:id="echoid-s12822" xml:space="preserve">: R x FZ. </s>
  <s xml:id="echoid-s12823" xml:space="preserve">FZq:</s>
  <s xml:id="echoid-s12824" xml:space="preserve">: R x EY. <lb/></s>
  <s xml:id="echoid-s12825" xml:space="preserve">FZq. </s>
  <s xml:id="echoid-s12826" xml:space="preserve">adeóque FG x FZq = ES x R x EY.</s>
  <s xml:id="echoid-s12827" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12828" xml:space="preserve">XXI. </s>
  <s xml:id="echoid-s12829" xml:space="preserve">Simili ratione _ſumma Cuborum_ ex FZ æquatur ei quod fit ex R <lb/>in ſummam quadratorum ex rectis EY ad BD applicatis. </s>
  <s xml:id="echoid-s12830" xml:space="preserve">neque non ſi-<lb/>mili quoad reliquas poteſtates tenore.</s>
  <s xml:id="echoid-s12831" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12832" xml:space="preserve">XXII. </s>
  <s xml:id="echoid-s12833" xml:space="preserve">Sit curva quævis DOK, in qua deſignatum punctum D; <lb/></s>
  <s xml:id="echoid-s12834" xml:space="preserve">
<anchor type="note" xlink:label="note-0269-01a" xlink:href="note-0269-01"/>
&amp; </s>
  <s xml:id="echoid-s12835" xml:space="preserve">ſubtenſa recta DK; </s>
  <s xml:id="echoid-s12836" xml:space="preserve">ſit item curva AE talis, ut à D projectâ quâ-<lb/>vis rectâ DMF (quæ curvas ſecet punctis M, F) ductíſque DS ad <lb/>DM normali, &amp; </s>
  <s xml:id="echoid-s12837" xml:space="preserve">MS curvam DOK tangente (concurrentibus utiq; <lb/></s>
  <s xml:id="echoid-s12838" xml:space="preserve">puncto S) datâque quâdam R, ſit DS. </s>
  <s xml:id="echoid-s12839" xml:space="preserve">2 R:</s>
  <s xml:id="echoid-s12840" xml:space="preserve">: DMq. </s>
  <s xml:id="echoid-s12841" xml:space="preserve">DFq; </s>
  <s xml:id="echoid-s12842" xml:space="preserve">erit <lb/>ſpatium ADE æquale ex R, DK.</s>
  <s xml:id="echoid-s12843" xml:space="preserve"/>
</p>
<div xml:id="echoid-div391" type="float" level="2" n="11">
<note position="right" xlink:label="note-0269-01" xlink:href="note-0269-01a" xml:space="preserve">Fig. 128.</note>
</div>
<p>
  <s xml:id="echoid-s12844" xml:space="preserve">Nam ſubtenſa DK indefinitè ſecta concipiatur punctis PQ, &amp;</s>
  <s xml:id="echoid-s12845" xml:space="preserve">c. <lb/></s>
  <s xml:id="echoid-s12846" xml:space="preserve">per quæ centro C deſcripti tranſeant arcus PM, QRN; </s>
  <s xml:id="echoid-s12847" xml:space="preserve">curvam <lb/>DOK ſecantes punctis M, N; </s>
  <s xml:id="echoid-s12848" xml:space="preserve">per quæ ducantur rectæ DMF, <lb/>DNG; </s>
  <s xml:id="echoid-s12849" xml:space="preserve">ſint verò DT ad DK; </s>
  <s xml:id="echoid-s12850" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12851" xml:space="preserve">DS ad DM perpendiculares; </s>
  <s xml:id="echoid-s12852" xml:space="preserve"><lb/>quibus occurrant tangentes KT, MS. </s>
  <s xml:id="echoid-s12853" xml:space="preserve">demùm centro D per E duca-<lb/>tur arcus EX; </s>
  <s xml:id="echoid-s12854" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12855" xml:space="preserve">per F arcus FY. </s>
  <s xml:id="echoid-s12856" xml:space="preserve">Jam, ob ſectionem indefinitam, <lb/>eſt triangulum KPM triangulo KDT ſimile. </s>
  <s xml:id="echoid-s12857" xml:space="preserve">ac ideò MP. </s>
  <s xml:id="echoid-s12858" xml:space="preserve">PK:</s>
  <s xml:id="echoid-s12859" xml:space="preserve">: <lb/>TD. </s>
  <s xml:id="echoid-s12860" xml:space="preserve">DK. </s>
  <s xml:id="echoid-s12861" xml:space="preserve">item eſt DP. </s>
  <s xml:id="echoid-s12862" xml:space="preserve">PM:</s>
  <s xml:id="echoid-s12863" xml:space="preserve">: DE. </s>
  <s xml:id="echoid-s12864" xml:space="preserve">EX. </s>
  <s xml:id="echoid-s12865" xml:space="preserve">ſeu, propter aſſigna-<lb/>tam cauſam, DK. </s>
  <s xml:id="echoid-s12866" xml:space="preserve">MP:</s>
  <s xml:id="echoid-s12867" xml:space="preserve">: DE. </s>
  <s xml:id="echoid-s12868" xml:space="preserve">EX. </s>
  <s xml:id="echoid-s12869" xml:space="preserve">Eſt itaque MP x DK. </s>
  <s xml:id="echoid-s12870" xml:space="preserve">PK x <lb/>MP:</s>
  <s xml:id="echoid-s12871" xml:space="preserve">: TD x DE. </s>
  <s xml:id="echoid-s12872" xml:space="preserve">DK x EX. </s>
  <s xml:id="echoid-s12873" xml:space="preserve">hoc eſt DK. </s>
  <s xml:id="echoid-s12874" xml:space="preserve">PK:</s>
  <s xml:id="echoid-s12875" xml:space="preserve">: TD x DEq. </s>
  <s xml:id="echoid-s12876" xml:space="preserve"><lb/>DK x EX x DE. </s>
  <s xml:id="echoid-s12877" xml:space="preserve">ac inde DKq x EX x DE = PK x TD x <lb/>DEq. </s>
  <s xml:id="echoid-s12878" xml:space="preserve">(_a_) Eſt autem DT. </s>
  <s xml:id="echoid-s12879" xml:space="preserve">2 R:</s>
  <s xml:id="echoid-s12880" xml:space="preserve">: DKq. </s>
  <s xml:id="echoid-s12881" xml:space="preserve">DEq; </s>
  <s xml:id="echoid-s12882" xml:space="preserve">ſeu DT x DEq <lb/>
<anchor type="note" xlink:label="note-0269-02a" xlink:href="note-0269-02"/>
= 2 R x DKq. </s>
  <s xml:id="echoid-s12883" xml:space="preserve">ergò eſt DKq x EX x DE = PK x 2 R x DKq. <lb/></s>
  <s xml:id="echoid-s12884" xml:space="preserve">quare EX x DE = 2 R x PK; </s>
  <s xml:id="echoid-s12885" xml:space="preserve">hoc eſt, 2 ſector DEX = 2 R x PK. </s>
  <s xml:id="echoid-s12886" xml:space="preserve"><lb/>unde ſector DEX = R x PK. </s>
  <s xml:id="echoid-s12887" xml:space="preserve">Simili planè diſcurſu ſector DFY
<pb o="92" file="0270" n="285" rhead=""/>
a<unsure/>quatur ipſi R x RM, vel R x QP. </s>
  <s xml:id="echoid-s12888" xml:space="preserve">itaque totum ſpatium ADE <lb/>quod ab ejuſmodi ſectoribus minimè differt adæquatur toti R x DK. <lb/></s>
  <s xml:id="echoid-s12889" xml:space="preserve">quod erat Propoſitum.</s>
  <s xml:id="echoid-s12890" xml:space="preserve"/>
</p>
<div xml:id="echoid-div392" type="float" level="2" n="12">
<note position="right" xlink:label="note-0269-02" xlink:href="note-0269-02a" xml:space="preserve">(_a_) _Hyp._</note>
</div>
<p>
  <s xml:id="echoid-s12891" xml:space="preserve">XXIII. </s>
  <s xml:id="echoid-s12892" xml:space="preserve">Iiſdem, quoad cætera, poſitis atque paratis, ducantur KH <lb/>
<anchor type="note" xlink:label="note-0270-01a" xlink:href="note-0270-01"/>
ad KT, &amp; </s>
  <s xml:id="echoid-s12893" xml:space="preserve">MI ad MS perpendiculares; </s>
  <s xml:id="echoid-s12894" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12895" xml:space="preserve">concipiatur jam curva <lb/>AE naturâ talis, ut ſit DE = √ DK x DH; </s>
  <s xml:id="echoid-s12896" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12897" xml:space="preserve">DF = √ DM x <lb/>DI; </s>
  <s xml:id="echoid-s12898" xml:space="preserve">ac ità perpetuò; </s>
  <s xml:id="echoid-s12899" xml:space="preserve">erit ſpatium ADE quadrati ex DK ſubqua-<lb/>druplum.</s>
  <s xml:id="echoid-s12900" xml:space="preserve"/>
</p>
<div xml:id="echoid-div393" type="float" level="2" n="13">
<note position="left" xlink:label="note-0270-01" xlink:href="note-0270-01a" xml:space="preserve">Fig. 128.</note>
</div>
<p>
  <s xml:id="echoid-s12901" xml:space="preserve">Nam eſt MP. </s>
  <s xml:id="echoid-s12902" xml:space="preserve">PK:</s>
  <s xml:id="echoid-s12903" xml:space="preserve">: DK. </s>
  <s xml:id="echoid-s12904" xml:space="preserve">DH:</s>
  <s xml:id="echoid-s12905" xml:space="preserve">: DKq. </s>
  <s xml:id="echoid-s12906" xml:space="preserve">DK x DH:</s>
  <s xml:id="echoid-s12907" xml:space="preserve">: DKq. <lb/></s>
  <s xml:id="echoid-s12908" xml:space="preserve">DEq. </s>
  <s xml:id="echoid-s12909" xml:space="preserve">item DP. </s>
  <s xml:id="echoid-s12910" xml:space="preserve">PM:</s>
  <s xml:id="echoid-s12911" xml:space="preserve">: DE. </s>
  <s xml:id="echoid-s12912" xml:space="preserve">EX; </s>
  <s xml:id="echoid-s12913" xml:space="preserve">hoc eſt DK. </s>
  <s xml:id="echoid-s12914" xml:space="preserve">PM:</s>
  <s xml:id="echoid-s12915" xml:space="preserve">: DE. </s>
  <s xml:id="echoid-s12916" xml:space="preserve"><lb/>EX. </s>
  <s xml:id="echoid-s12917" xml:space="preserve">ergò MP x DK. </s>
  <s xml:id="echoid-s12918" xml:space="preserve">PK x PM:</s>
  <s xml:id="echoid-s12919" xml:space="preserve">: DKq x DE. </s>
  <s xml:id="echoid-s12920" xml:space="preserve">DEq x EX. </s>
  <s xml:id="echoid-s12921" xml:space="preserve"><lb/>hoc eſt DK PK:</s>
  <s xml:id="echoid-s12922" xml:space="preserve">: DKq. </s>
  <s xml:id="echoid-s12923" xml:space="preserve">DE x EX. </s>
  <s xml:id="echoid-s12924" xml:space="preserve">vel DKq. </s>
  <s xml:id="echoid-s12925" xml:space="preserve">DK x PK:</s>
  <s xml:id="echoid-s12926" xml:space="preserve">: DKq. </s>
  <s xml:id="echoid-s12927" xml:space="preserve"><lb/>DE x EX. </s>
  <s xml:id="echoid-s12928" xml:space="preserve">unde DK x PK = DE x EX. </s>
  <s xml:id="echoid-s12929" xml:space="preserve">Simili ratione DM x MR <lb/>(vel DP x PQ) = DF x FY. </s>
  <s xml:id="echoid-s12930" xml:space="preserve">Verúm omnia DK x PK, DP x <lb/>PQ, &amp;</s>
  <s xml:id="echoid-s12931" xml:space="preserve">c æquantur ſemiſſi quadrati ex DK; </s>
  <s xml:id="echoid-s12932" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12933" xml:space="preserve">omnia DE x EX, <lb/>DF x FY, &amp;</s>
  <s xml:id="echoid-s12934" xml:space="preserve">c æquantur _duplo ſpatio_ EDA; </s>
  <s xml:id="echoid-s12935" xml:space="preserve">unde manifeſte con-<lb/>ſequitur Propoſitum.</s>
  <s xml:id="echoid-s12936" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12937" xml:space="preserve">XXIV. </s>
  <s xml:id="echoid-s12938" xml:space="preserve">Sit curva quæpiam DOK, in qua punctum D; </s>
  <s xml:id="echoid-s12939" xml:space="preserve">cuique <lb/>
<anchor type="note" xlink:label="note-0270-02a" xlink:href="note-0270-02"/>
ſubtendatur recta DK; </s>
  <s xml:id="echoid-s12940" xml:space="preserve">ſit item curva DZI talis, ut ſumpto in curva <lb/>DOK puncto quopiam M, connexâque DM; </s>
  <s xml:id="echoid-s12941" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12942" xml:space="preserve">ductâ DS ad DM <lb/>perpendiculari, &amp; </s>
  <s xml:id="echoid-s12943" xml:space="preserve">MS curvam DOK tangente; </s>
  <s xml:id="echoid-s12944" xml:space="preserve">ſumptâ demum <lb/>DP = DM, &amp; </s>
  <s xml:id="echoid-s12945" xml:space="preserve">ductâ PZ ad DK perpendiculari, ſit PZ = DS; <lb/></s>
  <s xml:id="echoid-s12946" xml:space="preserve">erit _ſpatium_ DKI æquale _duplo ſpatio_ DKOD.</s>
  <s xml:id="echoid-s12947" xml:space="preserve"/>
</p>
<div xml:id="echoid-div394" type="float" level="2" n="14">
<note position="left" xlink:label="note-0270-02" xlink:href="note-0270-02a" xml:space="preserve">Fig. 129.</note>
</div>
<p>
  <s xml:id="echoid-s12948" xml:space="preserve">Nam recta KP concipiatur indefinitè parva; </s>
  <s xml:id="echoid-s12949" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12950" xml:space="preserve">DT ipſi DK per-<lb/>pendicularis ſit, &amp; </s>
  <s xml:id="echoid-s12951" xml:space="preserve">KT curvam DOK tangat. </s>
  <s xml:id="echoid-s12952" xml:space="preserve">Eſt itaque (ducto <lb/>arcu MP) rurſus KP. </s>
  <s xml:id="echoid-s12953" xml:space="preserve">PM:</s>
  <s xml:id="echoid-s12954" xml:space="preserve">: KD. </s>
  <s xml:id="echoid-s12955" xml:space="preserve">DT:</s>
  <s xml:id="echoid-s12956" xml:space="preserve">: KD. </s>
  <s xml:id="echoid-s12957" xml:space="preserve">KI. </s>
  <s xml:id="echoid-s12958" xml:space="preserve">unde KP x <lb/>KI = PM x KD. </s>
  <s xml:id="echoid-s12959" xml:space="preserve">Capiatur alia particula PQ, &amp; </s>
  <s xml:id="echoid-s12960" xml:space="preserve">centro D per <lb/>Q ducatur arcus QN, quem ſecet ſubtenſa DM in R; </s>
  <s xml:id="echoid-s12961" xml:space="preserve">eſt ergòrur-<lb/>ſus MR. </s>
  <s xml:id="echoid-s12962" xml:space="preserve">RN:</s>
  <s xml:id="echoid-s12963" xml:space="preserve">: MD. </s>
  <s xml:id="echoid-s12964" xml:space="preserve">DS; </s>
  <s xml:id="echoid-s12965" xml:space="preserve">hoc eſt PQ. </s>
  <s xml:id="echoid-s12966" xml:space="preserve">RN:</s>
  <s xml:id="echoid-s12967" xml:space="preserve">: MD. </s>
  <s xml:id="echoid-s12968" xml:space="preserve">PZ qua-<lb/>re PQ x PZ = RN x MD; </s>
  <s xml:id="echoid-s12969" xml:space="preserve">ac ità continuò deinceps. </s>
  <s xml:id="echoid-s12970" xml:space="preserve">patet igitur <lb/>omnia ſimul rectangula KP x KI, PQ x PZ, &amp;</s>
  <s xml:id="echoid-s12971" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12972" xml:space="preserve">æquari aggrega-<lb/>to omnium PM x KD, RN x MD, &amp;</s>
  <s xml:id="echoid-s12973" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12974" xml:space="preserve">hoc eſt ſpatium DKI <lb/>duplo ſpatio DKOD æquari.</s>
  <s xml:id="echoid-s12975" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s12976" xml:space="preserve">XXV. </s>
  <s xml:id="echoid-s12977" xml:space="preserve">Iiſdem quoad cætera poſitis atque paratis, ordinatæ PZ jam <lb/>æquales concipiantur ipſis MS reſpectivis; </s>
  <s xml:id="echoid-s12978" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s12979" xml:space="preserve">ad rectam aſſumptam <lb/>
<anchor type="note" xlink:label="note-0270-03a" xlink:href="note-0270-03"/>
X_k_, diſtantiáſque X_k_, X_m_, X_n_, &amp;</s>
  <s xml:id="echoid-s12980" xml:space="preserve">c, æquales ipſis curvæ partibus <lb/>DOK, DOM, DON, &amp;</s>
  <s xml:id="echoid-s12981" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12982" xml:space="preserve">applicentur rectæ _kd_, _md_, _nd_, &amp;</s>
  <s xml:id="echoid-s12983" xml:space="preserve">c.</s>
  <s xml:id="echoid-s12984" xml:space="preserve">
<pb o="93" file="0271" n="286" rhead=""/>
pares ſubtenſis KD, MD, ND; </s>
  <s xml:id="echoid-s12985" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s12986" xml:space="preserve">c. </s>
  <s xml:id="echoid-s12987" xml:space="preserve">erit ſpatium X _k d_ æquale ſpa-<lb/>tio DKI.</s>
  <s xml:id="echoid-s12988" xml:space="preserve"/>
</p>
<div xml:id="echoid-div395" type="float" level="2" n="15">
<note position="left" xlink:label="note-0270-03" xlink:href="note-0270-03a" xml:space="preserve">Fig. 130.</note>
</div>
<p>
  <s xml:id="echoid-s12989" xml:space="preserve">Nam eſt KM. </s>
  <s xml:id="echoid-s12990" xml:space="preserve">KP:</s>
  <s xml:id="echoid-s12991" xml:space="preserve">: KT. </s>
  <s xml:id="echoid-s12992" xml:space="preserve">KD; </s>
  <s xml:id="echoid-s12993" xml:space="preserve">hoc eſt _km_. </s>
  <s xml:id="echoid-s12994" xml:space="preserve">KP:</s>
  <s xml:id="echoid-s12995" xml:space="preserve">: KI _kd_. <lb/></s>
  <s xml:id="echoid-s12996" xml:space="preserve">unde _km x k d_ = KP x KI. </s>
  <s xml:id="echoid-s12997" xml:space="preserve">Simiſique pacto, MN. </s>
  <s xml:id="echoid-s12998" xml:space="preserve">MR:</s>
  <s xml:id="echoid-s12999" xml:space="preserve">: MS. </s>
  <s xml:id="echoid-s13000" xml:space="preserve"><lb/>MD. </s>
  <s xml:id="echoid-s13001" xml:space="preserve">ſeu _mn_. </s>
  <s xml:id="echoid-s13002" xml:space="preserve">PQ:</s>
  <s xml:id="echoid-s13003" xml:space="preserve">: PZ. </s>
  <s xml:id="echoid-s13004" xml:space="preserve">_md_. </s>
  <s xml:id="echoid-s13005" xml:space="preserve">unde _mnx_ = PQ x PZ. </s>
  <s xml:id="echoid-s13006" xml:space="preserve"><lb/>ac ità deinceps. </s>
  <s xml:id="echoid-s13007" xml:space="preserve">unde cònſtat Propoſitum.</s>
  <s xml:id="echoid-s13008" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13009" xml:space="preserve">XXVI. </s>
  <s xml:id="echoid-s13010" xml:space="preserve">Sin porrò, perſiſtentibus reliquis, adſumptâ quâvis rectâ. <lb/></s>
  <s xml:id="echoid-s13011" xml:space="preserve">_kg_, completóque rectangulo X _kgb_, curva DZI talis intelligatur, <lb/>ut ſit MD. </s>
  <s xml:id="echoid-s13012" xml:space="preserve">MS:</s>
  <s xml:id="echoid-s13013" xml:space="preserve">: _k g_. </s>
  <s xml:id="echoid-s13014" xml:space="preserve">PZ; </s>
  <s xml:id="echoid-s13015" xml:space="preserve">erit rectangulum X _k g b_ æquale ſpatio <lb/>
<anchor type="note" xlink:label="note-0271-01a" xlink:href="note-0271-01"/>
DKI.</s>
  <s xml:id="echoid-s13016" xml:space="preserve"/>
</p>
<div xml:id="echoid-div396" type="float" level="2" n="16">
<note position="right" xlink:label="note-0271-01" xlink:href="note-0271-01a" xml:space="preserve">Fig. 130.</note>
</div>
<p>
  <s xml:id="echoid-s13017" xml:space="preserve">Nam eſt rurſus KP. </s>
  <s xml:id="echoid-s13018" xml:space="preserve">KM:</s>
  <s xml:id="echoid-s13019" xml:space="preserve">: KD. </s>
  <s xml:id="echoid-s13020" xml:space="preserve">KT:</s>
  <s xml:id="echoid-s13021" xml:space="preserve">: _k g_. </s>
  <s xml:id="echoid-s13022" xml:space="preserve">KI. </s>
  <s xml:id="echoid-s13023" xml:space="preserve">adeóque KP x <lb/>KI = (KM x _kg_ = ) _km_ x _kg_. </s>
  <s xml:id="echoid-s13024" xml:space="preserve">Similitérque PQ x PZ = _mn_ x <lb/>_kg_. </s>
  <s xml:id="echoid-s13025" xml:space="preserve">ac ità ſemper. </s>
  <s xml:id="echoid-s13026" xml:space="preserve">Unde conſtat.</s>
  <s xml:id="echoid-s13027" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13028" xml:space="preserve">Hinc noto ſpatio DKI cognoſcetur quantitas curvæ DOK.</s>
  <s xml:id="echoid-s13029" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13030" xml:space="preserve">Hujuſmodi verò complura deprehendet quiſquis hanc _Mineram_ pe-<lb/>nitiùs explorârit, ac excuſſerit. </s>
  <s xml:id="echoid-s13031" xml:space="preserve">Faciat cui id vacat &amp; </s>
  <s xml:id="echoid-s13032" xml:space="preserve">adlubeſ-<lb/>cit</s>
</p>
<p>
  <s xml:id="echoid-s13033" xml:space="preserve">XXVII. </s>
  <s xml:id="echoid-s13034" xml:space="preserve">Uſui fortè nonnunquam erit (mihi ſubinde fuit) &amp; </s>
  <s xml:id="echoid-s13035" xml:space="preserve">hoc, <lb/>è præmiſſis deductum Theorema.</s>
  <s xml:id="echoid-s13036" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 131.</note>
<p>
  <s xml:id="echoid-s13037" xml:space="preserve">Sit curva quæpiam VEH (cujus axis VD, baſis DH) quam tangat ut-<lb/>cunque recta ET; </s>
  <s xml:id="echoid-s13038" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13039" xml:space="preserve">ducatur EA ad HD parallela. </s>
  <s xml:id="echoid-s13040" xml:space="preserve">tum altera ſta-<lb/>tuatur curva GZZ talis, ut à puncto E ductâ EZ ad VD pa-<lb/>rallelâ (quæ baſin DH in I, curvam GZZ in Z ſecet) adſumptâq; <lb/></s>
  <s xml:id="echoid-s13041" xml:space="preserve">quâpiam determinatâ R, ſit ſemper DA q. </s>
  <s xml:id="echoid-s13042" xml:space="preserve">R q:</s>
  <s xml:id="echoid-s13043" xml:space="preserve">: DT. </s>
  <s xml:id="echoid-s13044" xml:space="preserve">IZ; </s>
  <s xml:id="echoid-s13045" xml:space="preserve">erit <lb/>DA. </s>
  <s xml:id="echoid-s13046" xml:space="preserve">AE:</s>
  <s xml:id="echoid-s13047" xml:space="preserve">: R q ſpat. </s>
  <s xml:id="echoid-s13048" xml:space="preserve">DIZG. </s>
  <s xml:id="echoid-s13049" xml:space="preserve">(vel facto DA. </s>
  <s xml:id="echoid-s13050" xml:space="preserve">R:</s>
  <s xml:id="echoid-s13051" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s13052" xml:space="preserve">DP; </s>
  <s xml:id="echoid-s13053" xml:space="preserve"><lb/>ductâque PQ ad DH parallelâ, erit _Rectangulum_ DPQI par _ſpa-_ <lb/>_tio_ DGZI).</s>
  <s xml:id="echoid-s13054" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13055" xml:space="preserve">Etiam hoc adjiciatur _Theorema;_ </s>
  <s xml:id="echoid-s13056" xml:space="preserve">nonnunquam uſui futurum.</s>
  <s xml:id="echoid-s13057" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13058" xml:space="preserve">XXVIII. </s>
  <s xml:id="echoid-s13059" xml:space="preserve">Sit curva quælibet AMB (cujus axis A D); </s>
  <s xml:id="echoid-s13060" xml:space="preserve">ſit item li-<lb/>
<anchor type="note" xlink:label="note-0271-03a" xlink:href="note-0271-03"/>
nea KZL proprietate talis, ut ſumpto in AMB quocunque puncto <lb/>M, &amp; </s>
  <s xml:id="echoid-s13061" xml:space="preserve">ab eo ductis rectâ MP ad curvam AB perpendiculari (quæ <lb/>axem AD ſecet in P) &amp; </s>
  <s xml:id="echoid-s13062" xml:space="preserve">rectà MG ad AD perpendiculari (quæ <lb/>curvam KZL ſecet in Z) ſit conſtantèr GM. </s>
  <s xml:id="echoid-s13063" xml:space="preserve">MP:</s>
  <s xml:id="echoid-s13064" xml:space="preserve">: arc AM. <lb/></s>
  <s xml:id="echoid-s13065" xml:space="preserve">GZ; </s>
  <s xml:id="echoid-s13066" xml:space="preserve">erit _ſpatium_ ADKL æquale _ſemiſſi quadrati_ ex arcn AM.</s>
  <s xml:id="echoid-s13067" xml:space="preserve"/>
</p>
<div xml:id="echoid-div397" type="float" level="2" n="17">
<note position="right" xlink:label="note-0271-03" xlink:href="note-0271-03a" xml:space="preserve">Fig. 132.</note>
</div>
<p>
  <s xml:id="echoid-s13068" xml:space="preserve">Hæcinquam, è præcedentibus haud magnâ o perâ colligantur, id <lb/>verò ſufficiat admonitum; </s>
  <s xml:id="echoid-s13069" xml:space="preserve">etenim hic animus eſt paulo ſubſiſtere.</s>
  <s xml:id="echoid-s13070" xml:space="preserve"/>
</p>
<pb o="94" file="0272" n="287" rhead=""/>
</div>
<div xml:id="echoid-div399" type="section" level="1" n="42">
<head xml:id="echoid-head45" xml:space="preserve">APPENDICUL A.</head>
<p>
  <s xml:id="echoid-s13071" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13072" xml:space="preserve">Cum pridem ante plures annos illuſtris Viri, _Chriſtiani Hugenii,_ <lb/>_Cyclometrica_ luſtrarem, ac in eo verſatus adverterem ad id <lb/>negotii duas præſertim ab ipſo methodos adhiberi; </s>
  <s xml:id="echoid-s13073" xml:space="preserve">quarum una _Cir-_ <lb/>_culi ſegmentum_ duobus parabolicis (uni inſcripto, alteri adſcripto) <lb/>medium eſſe monſtrans, illius inde magnitudini limites præſcribit; <lb/></s>
  <s xml:id="echoid-s13074" xml:space="preserve">altera _Parabolici ſegmenti, &amp; </s>
  <s xml:id="echoid-s13075" xml:space="preserve">Parallelogrammi_ æquè altorum cen-<lb/>tris gravitatum medium interjacere centrum gravitatis circularis ſeg-<lb/>menti oſtendens, alteros exindè limites, adſignat; </s>
  <s xml:id="echoid-s13076" xml:space="preserve">incidit mihi cogi-<lb/>tatio poſſe loco parabolæ in prima methodo, nec non vice Parallelo-<lb/>grammi in ſecunda, paraboliformium aliquam circulari ſegmento cir-<lb/>cumſcriptibilem uſurpari, ſic ut res aliquanto propiùs attingatur; </s>
  <s xml:id="echoid-s13077" xml:space="preserve">id <lb/>mox verum eſſe re perpensâ comperi; </s>
  <s xml:id="echoid-s13078" xml:space="preserve">quin&amp;</s>
  <s xml:id="echoid-s13079" xml:space="preserve">prætereà notavi facilèſup-<lb/>pares methodos _Hyperbolici ſegmenti dimenſioni_ accommodari. </s>
  <s xml:id="echoid-s13080" xml:space="preserve">Quo-<lb/>rum demonſtratio (præ aliis fortaſſe, quæ excogitari poſſent) brevis <lb/>&amp; </s>
  <s xml:id="echoid-s13081" xml:space="preserve">clara cùm è ſuprà poſitis conſequatur aut pendeat, eam (alioquin <lb/>opinor haud injucundam) hîc viſum eſt apponere.</s>
  <s xml:id="echoid-s13082" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13083" xml:space="preserve">II. </s>
  <s xml:id="echoid-s13084" xml:space="preserve">Adſumimus autem hæc pervulgata; </s>
  <s xml:id="echoid-s13085" xml:space="preserve">quorúmque demonſtratio-<lb/>nes è præmonſtratis haud difficilè variis modis colligantur; </s>
  <s xml:id="echoid-s13086" xml:space="preserve">ſi _parabo-_ <lb/>
<anchor type="note" xlink:label="note-0272-01a" xlink:href="note-0272-01"/>
_liformis_ BAE (cujus _Axis_ AD, _Baſis_ vel ordinata BDE, _Tan-_ <lb/>_gens_ BT; </s>
  <s xml:id="echoid-s13087" xml:space="preserve">_Gravitatis centrum_ K) exponens ſit {_n_/_m_}; </s>
  <s xml:id="echoid-s13088" xml:space="preserve">erit _Area_ BAE <lb/> = {_m_/_n_ + _m_} AD x BE; </s>
  <s xml:id="echoid-s13089" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13090" xml:space="preserve">TD = {_m_/_n_} AD, &amp; </s>
  <s xml:id="echoid-s13091" xml:space="preserve">KD = {_m_/_n_ + 2 _m_} AD.</s>
  <s xml:id="echoid-s13092" xml:space="preserve"/>
</p>
<div xml:id="echoid-div399" type="float" level="2" n="1">
<note position="left" xlink:label="note-0272-01" xlink:href="note-0272-01a" xml:space="preserve">Fig. 133.</note>
</div>
<p>
  <s xml:id="echoid-s13093" xml:space="preserve">III. </s>
  <s xml:id="echoid-s13094" xml:space="preserve">Sint duæ quævis curvæ AEB, AFB (quarum communis <lb/>
<anchor type="note" xlink:label="note-0272-02a" xlink:href="note-0272-02"/>
axis AD, ordinata DB) ità ſe habentes, ut ductâ quâcunque rectâ <lb/>EFG ad BD parallelâ, quæ lineas expoſitas punctis E, F, G ſecet, po-<lb/>ſitóque quòd rectæ ES, FT tangant curvas, (illa curvam AEB, hæc
<pb o="95" file="0273" n="288" rhead=""/>
ipſam AFB) ſit perpetuo TG major quàm SG; </s>
  <s xml:id="echoid-s13095" xml:space="preserve">dico nullam cur-<lb/>væ AFB partem intra ipſam AEB cadere.</s>
  <s xml:id="echoid-s13096" xml:space="preserve"/>
</p>
<div xml:id="echoid-div400" type="float" level="2" n="2">
<note position="left" xlink:label="note-0272-02" xlink:href="note-0272-02a" xml:space="preserve">Fig. 134.</note>
</div>
<p>
  <s xml:id="echoid-s13097" xml:space="preserve">Si fieri poteſt, cadat pars NFM; </s>
  <s xml:id="echoid-s13098" xml:space="preserve">ità ſcilicet ut curva AFB cur-<lb/>vam AEB interſecet punctis M, N; </s>
  <s xml:id="echoid-s13099" xml:space="preserve">his autem interjecta concipiatur <lb/>indeterminatè ordinata EFG; </s>
  <s xml:id="echoid-s13100" xml:space="preserve">ſint verò lineæ LXK, RYQ tales, <lb/>utductis rectis EO, FP ad ipſas ES, FT perpendicularibus, protra-<lb/>ctâque rectâ EG, ut hæc dictas lineas LK, QR ſecet punctis X, Y; <lb/></s>
  <s xml:id="echoid-s13101" xml:space="preserve">ſit GX = GO, &amp; </s>
  <s xml:id="echoid-s13102" xml:space="preserve">GY = GP. </s>
  <s xml:id="echoid-s13103" xml:space="preserve">Jam ex oſtenſis patet eſſe _ſpatium_ <lb/>IHKL = {HMq - INq/2} = ſpat. </s>
  <s xml:id="echoid-s13104" xml:space="preserve">IHQR; </s>
  <s xml:id="echoid-s13105" xml:space="preserve">adeóq; </s>
  <s xml:id="echoid-s13106" xml:space="preserve">ſpat. </s>
  <s xml:id="echoid-s13107" xml:space="preserve">IHKL, IHQR <lb/>æquari. </s>
  <s xml:id="echoid-s13108" xml:space="preserve">Verùm ob GE. </s>
  <s xml:id="echoid-s13109" xml:space="preserve">GO (GX):</s>
  <s xml:id="echoid-s13110" xml:space="preserve">: SG; </s>
  <s xml:id="echoid-s13111" xml:space="preserve">GE. </s>
  <s xml:id="echoid-s13112" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s13113" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13114" xml:space="preserve">SG. </s>
  <s xml:id="echoid-s13115" xml:space="preserve">GF &amp;</s>
  <s xml:id="echoid-s13116" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13117" xml:space="preserve">TG. </s>
  <s xml:id="echoid-s13118" xml:space="preserve">GF:</s>
  <s xml:id="echoid-s13119" xml:space="preserve">: <lb/>GF. </s>
  <s xml:id="echoid-s13120" xml:space="preserve">GP (GY) &amp;</s>
  <s xml:id="echoid-s13121" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13122" xml:space="preserve">GE. </s>
  <s xml:id="echoid-s13123" xml:space="preserve">GY; </s>
  <s xml:id="echoid-s13124" xml:space="preserve">eſt GX &amp;</s>
  <s xml:id="echoid-s13125" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13126" xml:space="preserve">GY; </s>
  <s xml:id="echoid-s13127" xml:space="preserve">adeòque (cùm <lb/>hoc ubique ſimiliter contingat) ſpatium IHKL majus ſpatio IHQR; </s>
  <s xml:id="echoid-s13128" xml:space="preserve"><lb/>quod repugnat oſtenſo. </s>
  <s xml:id="echoid-s13129" xml:space="preserve">itaque liquet Propoſitum.</s>
  <s xml:id="echoid-s13130" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13131" xml:space="preserve">Hinc tota AFB extra totam AEB jacet, nec illa hane uſquam inter-<lb/>ſecat.</s>
  <s xml:id="echoid-s13132" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13133" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s13134" xml:space="preserve">Sit curva quæpiam BAE, cujus axis AD, &amp; </s>
  <s xml:id="echoid-s13135" xml:space="preserve">ad hunc ordina-<lb/>
<anchor type="note" xlink:label="note-0273-01a" xlink:href="note-0273-01"/>
ta baſis ADE; </s>
  <s xml:id="echoid-s13136" xml:space="preserve">ſegmenti verò BAE centrum gravitatis ſit punctum <lb/>H, qer quod ducta ſit recta RS ad BE parallela Porrò per puncta <lb/>R, S tranſeat altera curva (vel linea quævis) MR ASN, habens iti-<lb/>dem axin AD, ac ita priorem curvam BAE ſecans, ut ejuſce pars <lb/>ſuperior RKAP Sintra curvam BAE cadat, inferiores verò reliquæ <lb/>partes RM, SN extra eandem; </s>
  <s xml:id="echoid-s13137" xml:space="preserve">erit ſegmenti MRASN centrum <lb/>gravitatis infra punctum H, verſus baſin MN.</s>
  <s xml:id="echoid-s13138" xml:space="preserve"/>
</p>
<div xml:id="echoid-div401" type="float" level="2" n="3">
<note position="right" xlink:label="note-0273-01" xlink:href="note-0273-01a" xml:space="preserve">Fig. 135.</note>
</div>
<p>
  <s xml:id="echoid-s13139" xml:space="preserve">Nam è ſegmento RIAO Sablatum RIAK + AOSP reſidu-<lb/>um BRKAPSE deprimet verſus baſin BE, puta ut jam ſit hujus <lb/>reſidui _Centrum gravitatis_ ad X; </s>
  <s xml:id="echoid-s13140" xml:space="preserve">tunc adjunctum BRM + ESN <lb/>adhuc totum MRKAPSN magis deprimet; </s>
  <s xml:id="echoid-s13141" xml:space="preserve">adeóque centrum ejus <lb/>infra X conſiſtet, velut ad Y. </s>
  <s xml:id="echoid-s13142" xml:space="preserve">itaque conſtat Propoſitum.</s>
  <s xml:id="echoid-s13143" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13144" xml:space="preserve">V. </s>
  <s xml:id="echoid-s13145" xml:space="preserve">_Circulum_ AFB, cujus _Centrum_ C, tangant duæ rectæ BT, E S <lb/>
<anchor type="note" xlink:label="note-0273-02a" xlink:href="note-0273-02"/>
_Diametro_ CA occurrentes punctis T, S; </s>
  <s xml:id="echoid-s13146" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13147" xml:space="preserve">ad CA perpendiculares <lb/>ſint rectæ BD, EP; </s>
  <s xml:id="echoid-s13148" xml:space="preserve">ſit autem AD major quàm AP; </s>
  <s xml:id="echoid-s13149" xml:space="preserve">erit TD. </s>
  <s xml:id="echoid-s13150" xml:space="preserve">AD <lb/>&amp;</s>
  <s xml:id="echoid-s13151" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13152" xml:space="preserve">SP. </s>
  <s xml:id="echoid-s13153" xml:space="preserve">AP.</s>
  <s xml:id="echoid-s13154" xml:space="preserve"/>
</p>
<div xml:id="echoid-div402" type="float" level="2" n="4">
<note position="right" xlink:label="note-0273-02" xlink:href="note-0273-02a" xml:space="preserve">Fig. 136.</note>
</div>
<p>
  <s xml:id="echoid-s13155" xml:space="preserve">Nam eſt CT. </s>
  <s xml:id="echoid-s13156" xml:space="preserve">CA:</s>
  <s xml:id="echoid-s13157" xml:space="preserve">: CA. </s>
  <s xml:id="echoid-s13158" xml:space="preserve">CD. </s>
  <s xml:id="echoid-s13159" xml:space="preserve">Ideoque CT - CA. </s>
  <s xml:id="echoid-s13160" xml:space="preserve">CA -<lb/>CD:</s>
  <s xml:id="echoid-s13161" xml:space="preserve">: CT. </s>
  <s xml:id="echoid-s13162" xml:space="preserve">CA; </s>
  <s xml:id="echoid-s13163" xml:space="preserve">hoc eſt TA. </s>
  <s xml:id="echoid-s13164" xml:space="preserve">AD:</s>
  <s xml:id="echoid-s13165" xml:space="preserve">: CT. </s>
  <s xml:id="echoid-s13166" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s13167" xml:space="preserve">Simili ratione conſtabit <lb/>eſſe SA. </s>
  <s xml:id="echoid-s13168" xml:space="preserve">AP:</s>
  <s xml:id="echoid-s13169" xml:space="preserve">: CS. </s>
  <s xml:id="echoid-s13170" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s13171" xml:space="preserve">Eſt autem CT. </s>
  <s xml:id="echoid-s13172" xml:space="preserve">CA &amp;</s>
  <s xml:id="echoid-s13173" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13174" xml:space="preserve">CS. </s>
  <s xml:id="echoid-s13175" xml:space="preserve">CA. <lb/></s>
  <s xml:id="echoid-s13176" xml:space="preserve">quare TA, AD &amp;</s>
  <s xml:id="echoid-s13177" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13178" xml:space="preserve">SA. </s>
  <s xml:id="echoid-s13179" xml:space="preserve">AP. </s>
  <s xml:id="echoid-s13180" xml:space="preserve">vel componendo TD. </s>
  <s xml:id="echoid-s13181" xml:space="preserve">AD &amp;</s>
  <s xml:id="echoid-s13182" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13183" xml:space="preserve">SP. </s>
  <s xml:id="echoid-s13184" xml:space="preserve"><lb/>AP.</s>
  <s xml:id="echoid-s13185" xml:space="preserve"/>
</p>
<pb o="96" file="0274" n="289" rhead=""/>
<p>
  <s xml:id="echoid-s13186" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s13187" xml:space="preserve">_Hyperbolam_ AEB, cujus _Centrum_ C, tangant duæ rectæ <lb/>BT, ES, &amp; </s>
  <s xml:id="echoid-s13188" xml:space="preserve">reliqua ponantur ut in proximè præcedente; </s>
  <s xml:id="echoid-s13189" xml:space="preserve">erit T D: <lb/></s>
  <s xml:id="echoid-s13190" xml:space="preserve">
<anchor type="note" xlink:label="note-0274-01a" xlink:href="note-0274-01"/>
A D &amp;</s>
  <s xml:id="echoid-s13191" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13192" xml:space="preserve">SP. </s>
  <s xml:id="echoid-s13193" xml:space="preserve">AP.</s>
  <s xml:id="echoid-s13194" xml:space="preserve"/>
</p>
<div xml:id="echoid-div403" type="float" level="2" n="5">
<note position="left" xlink:label="note-0274-01" xlink:href="note-0274-01a" xml:space="preserve">Fig. 137.</note>
</div>
<p>
  <s xml:id="echoid-s13195" xml:space="preserve">Nam eſt CA. </s>
  <s xml:id="echoid-s13196" xml:space="preserve">CD:</s>
  <s xml:id="echoid-s13197" xml:space="preserve">: CT. </s>
  <s xml:id="echoid-s13198" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s13199" xml:space="preserve">unde CA - CT. </s>
  <s xml:id="echoid-s13200" xml:space="preserve">CD -<lb/>CA:</s>
  <s xml:id="echoid-s13201" xml:space="preserve">: CT. </s>
  <s xml:id="echoid-s13202" xml:space="preserve">CA; </s>
  <s xml:id="echoid-s13203" xml:space="preserve">hoceſt TA. </s>
  <s xml:id="echoid-s13204" xml:space="preserve">AD:</s>
  <s xml:id="echoid-s13205" xml:space="preserve">: CT. </s>
  <s xml:id="echoid-s13206" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s13207" xml:space="preserve">ſuppare diſ-<lb/>curſu, eſt SA. </s>
  <s xml:id="echoid-s13208" xml:space="preserve">AP:</s>
  <s xml:id="echoid-s13209" xml:space="preserve">: CS. </s>
  <s xml:id="echoid-s13210" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s13211" xml:space="preserve">Verùm eſt CT. </s>
  <s xml:id="echoid-s13212" xml:space="preserve">CA &amp;</s>
  <s xml:id="echoid-s13213" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13214" xml:space="preserve">CS. <lb/></s>
  <s xml:id="echoid-s13215" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s13216" xml:space="preserve">quare TA. </s>
  <s xml:id="echoid-s13217" xml:space="preserve">AD &amp;</s>
  <s xml:id="echoid-s13218" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13219" xml:space="preserve">SA. </s>
  <s xml:id="echoid-s13220" xml:space="preserve">AP; </s>
  <s xml:id="echoid-s13221" xml:space="preserve">ſeu componendo TD. </s>
  <s xml:id="echoid-s13222" xml:space="preserve">AD <lb/>&amp;</s>
  <s xml:id="echoid-s13223" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13224" xml:space="preserve">SP. </s>
  <s xml:id="echoid-s13225" xml:space="preserve">AP.</s>
  <s xml:id="echoid-s13226" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13227" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s13228" xml:space="preserve">_Circali_ AEB (cujus _Centrum_ C) &amp; </s>
  <s xml:id="echoid-s13229" xml:space="preserve">_paraboliformis_ AFB <lb/>communes ſint axis AD, &amp; </s>
  <s xml:id="echoid-s13230" xml:space="preserve">baſis BD; </s>
  <s xml:id="echoid-s13231" xml:space="preserve">ſit autem _paraboliformis_ ex-<lb/>ponens {_n_/_m_}; </s>
  <s xml:id="echoid-s13232" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13233" xml:space="preserve">AD = {_m_ - 2 _n_/_m_ - _n_} CA (vel _m_ - _n_. </s>
  <s xml:id="echoid-s13234" xml:space="preserve">_m_ - 2 _n_:</s>
  <s xml:id="echoid-s13235" xml:space="preserve">: <lb/>CA. </s>
  <s xml:id="echoid-s13236" xml:space="preserve">AD) _circulum_ verò tangat recta BT; </s>
  <s xml:id="echoid-s13237" xml:space="preserve">hæc quoque _paraboli-_ <lb/>_formem_ AFB continget.</s>
  <s xml:id="echoid-s13238" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13239" xml:space="preserve">Nam quia BT _circulum_ tangit, eſt CT CA:</s>
  <s xml:id="echoid-s13240" xml:space="preserve">: CA. </s>
  <s xml:id="echoid-s13241" xml:space="preserve">CD; </s>
  <s xml:id="echoid-s13242" xml:space="preserve">unde TA. <lb/></s>
  <s xml:id="echoid-s13243" xml:space="preserve">
<anchor type="note" xlink:label="note-0274-02a" xlink:href="note-0274-02"/>
AD:</s>
  <s xml:id="echoid-s13244" xml:space="preserve">:. </s>
  <s xml:id="echoid-s13245" xml:space="preserve">CACD.</s>
  <s xml:id="echoid-s13246" xml:space="preserve"><unsure/> componendóque TD. </s>
  <s xml:id="echoid-s13247" xml:space="preserve">AD:</s>
  <s xml:id="echoid-s13248" xml:space="preserve">: CA + CD. </s>
  <s xml:id="echoid-s13249" xml:space="preserve">CD Item, quo-<lb/>uiam eſt (ex hypotheſi) CA. </s>
  <s xml:id="echoid-s13250" xml:space="preserve">AD:</s>
  <s xml:id="echoid-s13251" xml:space="preserve">: _m_ - _n_. </s>
  <s xml:id="echoid-s13252" xml:space="preserve">_m_ -2 _n_; </s>
  <s xml:id="echoid-s13253" xml:space="preserve">erit per ratio-<lb/>nis converſionem CA. </s>
  <s xml:id="echoid-s13254" xml:space="preserve">CD:</s>
  <s xml:id="echoid-s13255" xml:space="preserve">: _m_ - _n. </s>
  <s xml:id="echoid-s13256" xml:space="preserve">n_. </s>
  <s xml:id="echoid-s13257" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13258" xml:space="preserve">componendo CA + <lb/>CD. </s>
  <s xml:id="echoid-s13259" xml:space="preserve">CD:</s>
  <s xml:id="echoid-s13260" xml:space="preserve">: _m. </s>
  <s xml:id="echoid-s13261" xml:space="preserve">n_. </s>
  <s xml:id="echoid-s13262" xml:space="preserve">hoc eſt TD. </s>
  <s xml:id="echoid-s13263" xml:space="preserve">AD:</s>
  <s xml:id="echoid-s13264" xml:space="preserve">: _m. </s>
  <s xml:id="echoid-s13265" xml:space="preserve">n_. </s>
  <s xml:id="echoid-s13266" xml:space="preserve">unde <anchor type="note" xlink:href="" symbol="(_a_)"/> palàm fit,
<anchor type="note" xlink:label="note-0274-03a" xlink:href="note-0274-03"/>
quòd BT _paraboliformem_ AFB tangit.</s>
  <s xml:id="echoid-s13267" xml:space="preserve"/>
</p>
<div xml:id="echoid-div404" type="float" level="2" n="6">
<note position="left" xlink:label="note-0274-02" xlink:href="note-0274-02a" xml:space="preserve">Fig. 138.</note>
<note symbol="(_a_)" position="left" xlink:label="note-0274-03" xlink:href="note-0274-03a" xml:space="preserve">2 _hujus ap._</note>
</div>
<p>
  <s xml:id="echoid-s13268" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s13269" xml:space="preserve">Subnotetur, quòd inversè, datâ ratione ipſius AD ad CA’ <lb/>deſignabitur hinc _paraboliformis_; </s>
  <s xml:id="echoid-s13270" xml:space="preserve">quæ _Circulum_ AEB ad B contin-<lb/>get. </s>
  <s xml:id="echoid-s13271" xml:space="preserve">Nempe, ſi AD = {_s_/_t_}, erit {_t_ - _s_/2 _t_ - _s_} dictæ _paraboliformis ex-_ <lb/>ponens. </s>
  <s xml:id="echoid-s13272" xml:space="preserve">Nam poſito fore {_t_ - _s_/2 _t_ - _s_} = {_n_/_m_}; </s>
  <s xml:id="echoid-s13273" xml:space="preserve">erit ideò (juxta crucem <lb/>multiplicando) _mt_ - _ms_ = 2 _tn_ - _sn_; </s>
  <s xml:id="echoid-s13274" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13275" xml:space="preserve">tranſponendo _mt_ -<lb/>2 _nt_ = _ms_ - _ns_. </s>
  <s xml:id="echoid-s13276" xml:space="preserve">ac ideò (æqualitatem ad analogiſmum redigendo) <lb/>_m_ - _n. </s>
  <s xml:id="echoid-s13277" xml:space="preserve">m_ - 2 _n_:</s>
  <s xml:id="echoid-s13278" xml:space="preserve">: _t. </s>
  <s xml:id="echoid-s13279" xml:space="preserve">s_:</s>
  <s xml:id="echoid-s13280" xml:space="preserve">: CA. </s>
  <s xml:id="echoid-s13281" xml:space="preserve">AD. </s>
  <s xml:id="echoid-s13282" xml:space="preserve">itaque conſtat ex anteceden-<lb/>te Propoſitum.</s>
  <s xml:id="echoid-s13283" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13284" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s13285" xml:space="preserve">Manente quoad cætera ſeptimæ hypotheſi, _paraboliformis_ <lb/>AFB extra _circulum_ AEB tota cadet.</s>
  <s xml:id="echoid-s13286" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13287" xml:space="preserve">Nam utcunque ducatur recta GEF ad DB parallela; </s>
  <s xml:id="echoid-s13288" xml:space="preserve">quæ ſecet <lb/>
<anchor type="note" xlink:label="note-0274-04a" xlink:href="note-0274-04"/>
circulum ad E, paraboliformem in F; </s>
  <s xml:id="echoid-s13289" xml:space="preserve">ductæque concipiantur rectæ <lb/>ES _circulum_, &amp; </s>
  <s xml:id="echoid-s13290" xml:space="preserve">recta FR _paraboliformem_ contingentes; </s>
  <s xml:id="echoid-s13291" xml:space="preserve">Eſtque
<pb o="97" file="0275" n="290" rhead=""/>
RG. </s>
  <s xml:id="echoid-s13292" xml:space="preserve">AG:</s>
  <s xml:id="echoid-s13293" xml:space="preserve">: <anchor type="note" xlink:href="" symbol="(_a_)"/> _m. </s>
  <s xml:id="echoid-s13294" xml:space="preserve">n_:</s>
  <s xml:id="echoid-s13295" xml:space="preserve">: TD. </s>
  <s xml:id="echoid-s13296" xml:space="preserve">AD <anchor type="note" xlink:href="" symbol="(_b_)"/> &amp;</s>
  <s xml:id="echoid-s13297" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13298" xml:space="preserve">SG. </s>
  <s xml:id="echoid-s13299" xml:space="preserve">AG. </s>
  <s xml:id="echoid-s13300" xml:space="preserve">quare RG
<anchor type="note" xlink:label="note-0275-01a" xlink:href="note-0275-01"/>
<anchor type="note" xlink:label="note-0275-02a" xlink:href="note-0275-02"/>
&amp;</s>
  <s xml:id="echoid-s13301" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13302" xml:space="preserve">SG. </s>
  <s xml:id="echoid-s13303" xml:space="preserve">unde <anchor type="note" xlink:href="" symbol="(_c_)"/> patet tota AFB extra circulum AEB jacere.</s>
  <s xml:id="echoid-s13304" xml:space="preserve"/>
</p>
<div xml:id="echoid-div405" type="float" level="2" n="7">
<note position="left" xlink:label="note-0274-04" xlink:href="note-0274-04a" xml:space="preserve">Fig. 139.</note>
<note symbol="(_a_)" position="right" xlink:label="note-0275-01" xlink:href="note-0275-01a" xml:space="preserve">2. _hujus._ <lb/>_app._</note>
<note symbol="(_b_)" position="right" xlink:label="note-0275-02" xlink:href="note-0275-02a" xml:space="preserve">5. _hujus ap._</note>
</div>
<note symbol="(_c_)" position="right" xml:space="preserve">3. _hujus ap._</note>
<p>
  <s xml:id="echoid-s13305" xml:space="preserve">X. </s>
  <s xml:id="echoid-s13306" xml:space="preserve">Reliquis itidem ſtantibus, ſiad baſin GE (utcunque parallelam <lb/>
<anchor type="note" xlink:label="note-0275-04a" xlink:href="note-0275-04"/>
ipſi DB) &amp; </s>
  <s xml:id="echoid-s13307" xml:space="preserve">axem AD conſtituta intelligatur _paraboliformis_ ejuſdem cum <lb/>ipſa AFB generis (nempe cujus etiam exponens {_n_/_m_}) illa ad partes A <lb/>ſupra GE, extra _circulum_ tota jacebit.</s>
  <s xml:id="echoid-s13308" xml:space="preserve"/>
</p>
<div xml:id="echoid-div406" type="float" level="2" n="8">
<note position="right" xlink:label="note-0275-04" xlink:href="note-0275-04a" xml:space="preserve">Fig. 139.</note>
</div>
<p>
  <s xml:id="echoid-s13309" xml:space="preserve">Nam in arcu AE accepto quocunque puncto M, ductâque MP ad <lb/>EG parallelâ, &amp; </s>
  <s xml:id="echoid-s13310" xml:space="preserve">MV circulum tangente; </s>
  <s xml:id="echoid-s13311" xml:space="preserve">eſt VP. </s>
  <s xml:id="echoid-s13312" xml:space="preserve">AP &amp;</s>
  <s xml:id="echoid-s13313" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13314" xml:space="preserve">SG. <lb/></s>
  <s xml:id="echoid-s13315" xml:space="preserve">AG &amp;</s>
  <s xml:id="echoid-s13316" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13317" xml:space="preserve">RG. </s>
  <s xml:id="echoid-s13318" xml:space="preserve">AG:</s>
  <s xml:id="echoid-s13319" xml:space="preserve">: _m. </s>
  <s xml:id="echoid-s13320" xml:space="preserve">n_; </s>
  <s xml:id="echoid-s13321" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_a_)"/> itaque rurſus liquet Propoſi-
<anchor type="note" xlink:label="note-0275-05a" xlink:href="note-0275-05"/>
tum.</s>
  <s xml:id="echoid-s13322" xml:space="preserve"/>
</p>
<div xml:id="echoid-div407" type="float" level="2" n="9">
<note symbol="(_a_)" position="right" xlink:label="note-0275-05" xlink:href="note-0275-05a" xml:space="preserve">3. _hujus ap._</note>
</div>
<p>
  <s xml:id="echoid-s13323" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s13324" xml:space="preserve">Conſectatur etiam dictam (ipſi AFB coordinatam &amp; </s>
  <s xml:id="echoid-s13325" xml:space="preserve">ad ba-<lb/>ſin GE conſtitutam) _paraboliformem_ infra GE ad DB protractam, <lb/>
<anchor type="note" xlink:label="note-0275-06a" xlink:href="note-0275-06"/>
eatenus intra _Circulum_ totam cadere,</s>
</p>
<div xml:id="echoid-div408" type="float" level="2" n="10">
<note position="right" xlink:label="note-0275-06" xlink:href="note-0275-06a" xml:space="preserve">Fig. 139.</note>
</div>
<p>
  <s xml:id="echoid-s13326" xml:space="preserve">Quòd intra _Circulum_ ſtatim infra EG cadet ex eo patet, quòd ipſam <lb/>tangens RE circulum ſecat (quia nempe SE circulum tangit). </s>
  <s xml:id="echoid-s13327" xml:space="preserve">quòd <lb/>alibi _Circulo_ non occurret hinc patet; </s>
  <s xml:id="echoid-s13328" xml:space="preserve">quoniam poſito quòd occurrat <lb/>uſpiam ad N, <anchor type="note" xlink:href="" symbol="(_a_)"/> tota ſupra N extra circulum caderet, contra quam
<anchor type="note" xlink:label="note-0275-07a" xlink:href="note-0275-07"/>
modò dictum ac oſtenſum eſt.</s>
  <s xml:id="echoid-s13329" xml:space="preserve"/>
</p>
<div xml:id="echoid-div409" type="float" level="2" n="11">
<note symbol="(_a_)" position="right" xlink:label="note-0275-07" xlink:href="note-0275-07a" xml:space="preserve">3. _hujus ap._</note>
</div>
<p>
  <s xml:id="echoid-s13330" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s13331" xml:space="preserve">Porrò, _Hyperbolæ_ AEB (cujus centrum C) &amp; </s>
  <s xml:id="echoid-s13332" xml:space="preserve">_parabolifor-_ <lb/>
<anchor type="note" xlink:label="note-0275-08a" xlink:href="note-0275-08"/>
_mis_ AFB, cujus exponens {_n_/_m_}, communes ſint axis AD, baſis DB; <lb/></s>
  <s xml:id="echoid-s13333" xml:space="preserve">ſit autem AD = {2 _n_ - _m_/_m_ - _n_} CA; </s>
  <s xml:id="echoid-s13334" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13335" xml:space="preserve">BT _hyperbolam_ tangat; </s>
  <s xml:id="echoid-s13336" xml:space="preserve">hæc <lb/>quoque _paraboliformem_ AFB continget.</s>
  <s xml:id="echoid-s13337" xml:space="preserve"/>
</p>
<div xml:id="echoid-div410" type="float" level="2" n="12">
<note position="right" xlink:label="note-0275-08" xlink:href="note-0275-08a" xml:space="preserve">Fig. 140.</note>
</div>
<p>
  <s xml:id="echoid-s13338" xml:space="preserve">Nam eſt CD. </s>
  <s xml:id="echoid-s13339" xml:space="preserve">CA:</s>
  <s xml:id="echoid-s13340" xml:space="preserve">: CA. </s>
  <s xml:id="echoid-s13341" xml:space="preserve">CT. </s>
  <s xml:id="echoid-s13342" xml:space="preserve">acindè AD. </s>
  <s xml:id="echoid-s13343" xml:space="preserve">TA:</s>
  <s xml:id="echoid-s13344" xml:space="preserve">: CD. </s>
  <s xml:id="echoid-s13345" xml:space="preserve">CA; </s>
  <s xml:id="echoid-s13346" xml:space="preserve">inverſéq; <lb/></s>
  <s xml:id="echoid-s13347" xml:space="preserve">componendo TD. </s>
  <s xml:id="echoid-s13348" xml:space="preserve">AD:</s>
  <s xml:id="echoid-s13349" xml:space="preserve">: CA + CD. </s>
  <s xml:id="echoid-s13350" xml:space="preserve">CD. </s>
  <s xml:id="echoid-s13351" xml:space="preserve">Verùm<unsure/> ex hypo-<lb/>theſi, eſt _m_ - _n_. </s>
  <s xml:id="echoid-s13352" xml:space="preserve">2 _n_ - _m_:</s>
  <s xml:id="echoid-s13353" xml:space="preserve">: CA. </s>
  <s xml:id="echoid-s13354" xml:space="preserve">CD; </s>
  <s xml:id="echoid-s13355" xml:space="preserve">adeoque inversè compo-<lb/>nendo CA. </s>
  <s xml:id="echoid-s13356" xml:space="preserve">CD:</s>
  <s xml:id="echoid-s13357" xml:space="preserve">: _m_ - _n. </s>
  <s xml:id="echoid-s13358" xml:space="preserve">n_: </s>
  <s xml:id="echoid-s13359" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13360" xml:space="preserve">rurſus componendo CA + C D. </s>
  <s xml:id="echoid-s13361" xml:space="preserve"><lb/>CD:</s>
  <s xml:id="echoid-s13362" xml:space="preserve">: _m. </s>
  <s xml:id="echoid-s13363" xml:space="preserve">n._ </s>
  <s xml:id="echoid-s13364" xml:space="preserve">hoc eſt TD. </s>
  <s xml:id="echoid-s13365" xml:space="preserve">AD:</s>
  <s xml:id="echoid-s13366" xml:space="preserve">: _m. </s>
  <s xml:id="echoid-s13367" xml:space="preserve">n_. </s>
  <s xml:id="echoid-s13368" xml:space="preserve">unde BT _hyperboliformem_ <lb/>contingit.</s>
  <s xml:id="echoid-s13369" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13370" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s13371" xml:space="preserve">Hinc rurſu datà ratione ipſius AD ad CA, _paraboliformis_ <lb/>ad punctum B _bype bolam_ contingens deſignabitur. </s>
  <s xml:id="echoid-s13372" xml:space="preserve">nempe ſit AD = <lb/>{_s_/_t_} CA; </s>
  <s xml:id="echoid-s13373" xml:space="preserve">erit {_n_/_m_} = {_t_ + _s_/2 _t_ + _s_}. </s>
  <s xml:id="echoid-s13374" xml:space="preserve">Nam hoc ſuppoſito erit (ςαυρδὸν
<pb o="98" file="0276" n="291" rhead=""/>
multiplicando) 2 _tn_ + _sn_ = _mt_ + _ms_. </s>
  <s xml:id="echoid-s13375" xml:space="preserve">vel tranſponendo 2 _nt_ -<lb/>_mt_ = _ms_ - _ns_. </s>
  <s xml:id="echoid-s13376" xml:space="preserve">unde _m_ - _n_. </s>
  <s xml:id="echoid-s13377" xml:space="preserve">2 _n_ - _m_:</s>
  <s xml:id="echoid-s13378" xml:space="preserve">: _t. </s>
  <s xml:id="echoid-s13379" xml:space="preserve">s_:</s>
  <s xml:id="echoid-s13380" xml:space="preserve">: CA. </s>
  <s xml:id="echoid-s13381" xml:space="preserve">AD. </s>
  <s xml:id="echoid-s13382" xml:space="preserve">er-<lb/>gò patet ex antecedente.</s>
  <s xml:id="echoid-s13383" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13384" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s13385" xml:space="preserve">Stante duodecimæ hypotheſi, _paraboliformis_ AFB intra hy-<lb/>perbolam AEB tota cadet.</s>
  <s xml:id="echoid-s13386" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13387" xml:space="preserve">Nam utcunque ducatur EFG ad BD parallela; </s>
  <s xml:id="echoid-s13388" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13389" xml:space="preserve">recta ER _hy-_ <lb/>
<anchor type="note" xlink:label="note-0276-01a" xlink:href="note-0276-01"/>
_perbolam_, recta FS _paraboliformem_ tangant. </s>
  <s xml:id="echoid-s13390" xml:space="preserve">Eſtque SG. </s>
  <s xml:id="echoid-s13391" xml:space="preserve">AG:</s>
  <s xml:id="echoid-s13392" xml:space="preserve">: <anchor type="note" xlink:href="" symbol="(_a_)"/>
<anchor type="note" xlink:label="note-0276-02a" xlink:href="note-0276-02"/>
_m. </s>
  <s xml:id="echoid-s13393" xml:space="preserve">n_:</s>
  <s xml:id="echoid-s13394" xml:space="preserve">: TD. </s>
  <s xml:id="echoid-s13395" xml:space="preserve">AD <anchor type="note" xlink:href="" symbol="(_b_)"/> &amp;</s>
  <s xml:id="echoid-s13396" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13397" xml:space="preserve">RG. </s>
  <s xml:id="echoid-s13398" xml:space="preserve">AG. </s>
  <s xml:id="echoid-s13399" xml:space="preserve">unde RG &amp;</s>
  <s xml:id="echoid-s13400" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13401" xml:space="preserve">SG. </s>
  <s xml:id="echoid-s13402" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_c_)"/> unde
<anchor type="note" xlink:label="note-0276-03a" xlink:href="note-0276-03"/>
<anchor type="note" xlink:label="note-0276-04a" xlink:href="note-0276-04"/>
curva AEB extra curvam AFB tota cadet.</s>
  <s xml:id="echoid-s13403" xml:space="preserve"/>
</p>
<div xml:id="echoid-div411" type="float" level="2" n="13">
<note position="left" xlink:label="note-0276-01" xlink:href="note-0276-01a" xml:space="preserve">Fig. 141.</note>
<note symbol="(_a_)" position="left" xlink:label="note-0276-02" xlink:href="note-0276-02a" xml:space="preserve">2. _hujus ap._</note>
<note symbol="(_b_)" position="left" xlink:label="note-0276-03" xlink:href="note-0276-03a" xml:space="preserve">6. _hujus ap._</note>
<note symbol="(_c_)" position="left" xlink:label="note-0276-04" xlink:href="note-0276-04a" xml:space="preserve">3. _hujus ap._</note>
</div>
<p>
  <s xml:id="echoid-s13404" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s13405" xml:space="preserve">Etiam, ſi reliquis perſtantibus, ad baſin GE, axin AG con-<lb/>ſtitutam imagineris ejuſdem ordinis _paraboliformem_; </s>
  <s xml:id="echoid-s13406" xml:space="preserve">hæc ad partes <lb/>
<anchor type="note" xlink:label="note-0276-05a" xlink:href="note-0276-05"/>
ipsâ GE ſuperiores intra _hyperbolam_ tota cadet.</s>
  <s xml:id="echoid-s13407" xml:space="preserve"/>
</p>
<div xml:id="echoid-div412" type="float" level="2" n="14">
<note position="left" xlink:label="note-0276-05" xlink:href="note-0276-05a" xml:space="preserve">Fig. 141.</note>
</div>
<p>
  <s xml:id="echoid-s13408" xml:space="preserve">Nam ſi in _curva hyperbolica_ AE ſumatur ubicunque punctum M, &amp; </s>
  <s xml:id="echoid-s13409" xml:space="preserve"><lb/>ordinetur MP, ducatúrque hyperbolam tangens MV; </s>
  <s xml:id="echoid-s13410" xml:space="preserve">erit VP. <lb/></s>
  <s xml:id="echoid-s13411" xml:space="preserve">AP &amp;</s>
  <s xml:id="echoid-s13412" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13413" xml:space="preserve">_m. </s>
  <s xml:id="echoid-s13414" xml:space="preserve">n._ </s>
  <s xml:id="echoid-s13415" xml:space="preserve">adeoque rurſus è tertia liquet Propoſitum.</s>
  <s xml:id="echoid-s13416" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13417" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s13418" xml:space="preserve">Quinetiam ſi hæc altera coordinata _paraboliformis_, ad baſin <lb/>EG conſtituta, ad DB protracta concipiatur, ejus ipſis EG, BD in-<lb/>
<anchor type="note" xlink:label="note-0276-06a" xlink:href="note-0276-06"/>
tercepta pars extra _hyperbolam_ tota cadet.</s>
  <s xml:id="echoid-s13419" xml:space="preserve"/>
</p>
<div xml:id="echoid-div413" type="float" level="2" n="15">
<note position="left" xlink:label="note-0276-06" xlink:href="note-0276-06a" xml:space="preserve">Fig. 141.</note>
</div>
<p>
  <s xml:id="echoid-s13420" xml:space="preserve">Nam quòd extra _hyperbolam_ infra EG cadit, exinde patet, quòd <lb/>ipſa cum ipſius tangente recta ES angulum efficit minorem eo, quem <lb/>eadem recta ES efficit cum recta RE hyperbolam tangente. </s>
  <s xml:id="echoid-s13421" xml:space="preserve">quòd au-<lb/>tem eadem alibi, velut ad N, _hyperbolæ_ non occurrit, patet; </s>
  <s xml:id="echoid-s13422" xml:space="preserve">quoniam <lb/>hoc poſito, <anchor type="note" xlink:href="" symbol="(_a_)"/> ipſa intra _hyperbolam_ AN tota conſiſteret, contra
<anchor type="note" xlink:label="note-0276-07a" xlink:href="note-0276-07"/>
quàm mox oſtenſum eſt.</s>
  <s xml:id="echoid-s13423" xml:space="preserve"/>
</p>
<div xml:id="echoid-div414" type="float" level="2" n="16">
<note symbol="(_a_)" position="left" xlink:label="note-0276-07" xlink:href="note-0276-07a" xml:space="preserve">3. _hujus ap._</note>
</div>
<p>
  <s xml:id="echoid-s13424" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s13425" xml:space="preserve">Habeant _Circulus_ AEB, &amp; </s>
  <s xml:id="echoid-s13426" xml:space="preserve">_parabola_ AFB communem <lb/>axem AD, &amp; </s>
  <s xml:id="echoid-s13427" xml:space="preserve">baſin DB; </s>
  <s xml:id="echoid-s13428" xml:space="preserve">_parabola_ ad partes ſupra BD intra _Circu-_ <lb/>_lum_; </s>
  <s xml:id="echoid-s13429" xml:space="preserve">at infra BD extra _circulum_ cadet.</s>
  <s xml:id="echoid-s13430" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13431" xml:space="preserve">Sit enim _Circuli Diameter_ AZ, &amp; </s>
  <s xml:id="echoid-s13432" xml:space="preserve">eiæqualis A Had BD paralle-<lb/>la, &amp; </s>
  <s xml:id="echoid-s13433" xml:space="preserve">connectatur ZH; </s>
  <s xml:id="echoid-s13434" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13435" xml:space="preserve">huic BD producta ad I; </s>
  <s xml:id="echoid-s13436" xml:space="preserve">ergo DI eſt <lb/>
<anchor type="note" xlink:label="note-0276-08a" xlink:href="note-0276-08"/>
_Parameter parabolæ_ AFB. </s>
  <s xml:id="echoid-s13437" xml:space="preserve">quòd ſi ſupra BD utcunque ducatur recta <lb/>EF GK ad BD parallela circulum ſecans in E, parabolam in F, rectas <lb/>AZ, HZ, in G, &amp; </s>
  <s xml:id="echoid-s13438" xml:space="preserve">K, patet eſſe GEq = AG x GK &amp;</s>
  <s xml:id="echoid-s13439" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13440" xml:space="preserve">AG x DI <lb/>= GFq. </s>
  <s xml:id="echoid-s13441" xml:space="preserve">unde GE &amp;</s>
  <s xml:id="echoid-s13442" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13443" xml:space="preserve">GF. </s>
  <s xml:id="echoid-s13444" xml:space="preserve">Item, ſi infra BD utcunque ducatur <lb/>recta MN OL ad BD parallela _parabolam_ ſecans in M, _circu-_ <lb/>_lum_ in N, rectas AZ, HZ in O, &amp; </s>
  <s xml:id="echoid-s13445" xml:space="preserve">L, itidem patet eſſe MO q <lb/>= AO x DI &amp;</s>
  <s xml:id="echoid-s13446" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13447" xml:space="preserve">AO x OL = NO q. </s>
  <s xml:id="echoid-s13448" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13449" xml:space="preserve">ideò M O
<pb o="99" file="0277" n="292" rhead=""/>
&amp;</s>
  <s xml:id="echoid-s13450" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13451" xml:space="preserve">NO. </s>
  <s xml:id="echoid-s13452" xml:space="preserve">quare liquent ea, quæ Propoſita ſunt.</s>
  <s xml:id="echoid-s13453" xml:space="preserve"/>
</p>
<div xml:id="echoid-div415" type="float" level="2" n="17">
<note position="left" xlink:label="note-0276-08" xlink:href="note-0276-08a" xml:space="preserve">Fig. 142.</note>
</div>
<p>
  <s xml:id="echoid-s13454" xml:space="preserve">Si _Circulo_ ſubſtituatur _Ellipſis_, eadem concluſio valet idem diſcur-<lb/>ſus probat; </s>
  <s xml:id="echoid-s13455" xml:space="preserve">pofitâ AH _Ellipſis parametro_.</s>
  <s xml:id="echoid-s13456" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13457" xml:space="preserve">XVIII Habeant _hyperbola_ AEB (cujus axis AZ, parameter AH) <lb/>&amp; </s>
  <s xml:id="echoid-s13458" xml:space="preserve">_parabola_ AFB axin eundem AD, &amp; </s>
  <s xml:id="echoid-s13459" xml:space="preserve">baſin DB, _parabola_ ſupra <lb/>DB tota extra _hyperbolam_ cadet, extra verò, ſi infra DB protraha-<lb/>
<anchor type="note" xlink:label="note-0277-01a" xlink:href="note-0277-01"/>
tur.</s>
  <s xml:id="echoid-s13460" xml:space="preserve"/>
</p>
<div xml:id="echoid-div416" type="float" level="2" n="18">
<note position="right" xlink:label="note-0277-01" xlink:href="note-0277-01a" xml:space="preserve">Fig. 143.</note>
</div>
<p>
  <s xml:id="echoid-s13461" xml:space="preserve">Nam connexæ ZH occurrat BD in I; </s>
  <s xml:id="echoid-s13462" xml:space="preserve">ergò DI eſt _parabolæ pa-_ <lb/>_rameter_. </s>
  <s xml:id="echoid-s13463" xml:space="preserve">Quòd ſi ſupra BD utcunque ducatur recta FEGK ad BD <lb/>parallela, ſecans hyperbolam in E, parabolam in F, rectas AD, ZH <lb/>punctis G, K, erit FGq = AG x DI &amp;</s>
  <s xml:id="echoid-s13464" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13465" xml:space="preserve">AG x GK = EGq. </s>
  <s xml:id="echoid-s13466" xml:space="preserve">qua-<lb/>re FG &amp;</s>
  <s xml:id="echoid-s13467" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13468" xml:space="preserve">EG. </s>
  <s xml:id="echoid-s13469" xml:space="preserve">Quòd ſiinfra BD, utcunque ducatur recta MNOL <lb/>ſecans hyperbolam in N, parabolam in M, rectas AD, ZH in O, &amp; </s>
  <s xml:id="echoid-s13470" xml:space="preserve"><lb/>L, erit NO q = AO x OL &amp;</s>
  <s xml:id="echoid-s13471" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13472" xml:space="preserve">AO x DI = MOq. </s>
  <s xml:id="echoid-s13473" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13474" xml:space="preserve">indè NO <lb/>&amp;</s>
  <s xml:id="echoid-s13475" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13476" xml:space="preserve">MO. </s>
  <s xml:id="echoid-s13477" xml:space="preserve">unde conſtant ea quæ propoſita ſunt.</s>
  <s xml:id="echoid-s13478" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13479" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s13480" xml:space="preserve">E dictis eliciuntur hæ _ad Circuli dimenſionem pertinentes regu-_ <lb/>_la._ </s>
  <s xml:id="echoid-s13481" xml:space="preserve">Sit BAE circuli portio, cujus axis AD, baſis BE; </s>
  <s xml:id="echoid-s13482" xml:space="preserve">ſitque C <lb/>
<anchor type="note" xlink:label="note-0277-02a" xlink:href="note-0277-02"/>
centrum circuli, &amp; </s>
  <s xml:id="echoid-s13483" xml:space="preserve">EH ſinus rectus arcus BAE; </s>
  <s xml:id="echoid-s13484" xml:space="preserve">item, ſit AD = <lb/>{_s_/_t_} CA; </s>
  <s xml:id="echoid-s13485" xml:space="preserve">erit 1. </s>
  <s xml:id="echoid-s13486" xml:space="preserve">{2 _t_ - _s_/3 _t_ - 2 _s_} AD x BE &amp;</s>
  <s xml:id="echoid-s13487" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13488" xml:space="preserve">port. </s>
  <s xml:id="echoid-s13489" xml:space="preserve">BAE.</s>
  <s xml:id="echoid-s13490" xml:space="preserve"/>
</p>
<div xml:id="echoid-div417" type="float" level="2" n="19">
<note position="right" xlink:label="note-0277-02" xlink:href="note-0277-02a" xml:space="preserve">Fig. 144.</note>
</div>
<p>
  <s xml:id="echoid-s13491" xml:space="preserve">2. </s>
  <s xml:id="echoid-s13492" xml:space="preserve">EH + {4 _t_ - 2 _s_/3 _t_ - 2 _s_} BH &amp;</s>
  <s xml:id="echoid-s13493" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13494" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s13495" xml:space="preserve">BAE.</s>
  <s xml:id="echoid-s13496" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13497" xml:space="preserve">3. </s>
  <s xml:id="echoid-s13498" xml:space="preserve">{2/3} AD x BE &amp;</s>
  <s xml:id="echoid-s13499" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13500" xml:space="preserve">port. </s>
  <s xml:id="echoid-s13501" xml:space="preserve">BAE.</s>
  <s xml:id="echoid-s13502" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13503" xml:space="preserve">4. </s>
  <s xml:id="echoid-s13504" xml:space="preserve">EH + {4/3} BH &amp;</s>
  <s xml:id="echoid-s13505" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13506" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s13507" xml:space="preserve">BAE.</s>
  <s xml:id="echoid-s13508" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13509" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s13510" xml:space="preserve">Itidem hæ deducuntur ad _hyperbolæ dimenſionem ſpectantes re-_ <lb/>_gulæ_. </s>
  <s xml:id="echoid-s13511" xml:space="preserve">Sit _hyperbolæ_ (cujus centrum C) ſegmentum ADB, habens <lb/>
<anchor type="note" xlink:label="note-0277-03a" xlink:href="note-0277-03"/>
axin AD = {_s_/_t_} CA; </s>
  <s xml:id="echoid-s13512" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13513" xml:space="preserve">baſin DB;</s>
  <s xml:id="echoid-s13514" xml:space="preserve"/>
</p>
<div xml:id="echoid-div418" type="float" level="2" n="20">
<note position="right" xlink:label="note-0277-03" xlink:href="note-0277-03a" xml:space="preserve">Fig. 145.</note>
</div>
<p>
  <s xml:id="echoid-s13515" xml:space="preserve">erit 1. </s>
  <s xml:id="echoid-s13516" xml:space="preserve">{2 _t_ + _s_/3 _t_ + 2 _s_} AD x DB &amp;</s>
  <s xml:id="echoid-s13517" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13518" xml:space="preserve">ſegm. </s>
  <s xml:id="echoid-s13519" xml:space="preserve">ADB. </s>
  <s xml:id="echoid-s13520" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s13521" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13522" xml:space="preserve">2. </s>
  <s xml:id="echoid-s13523" xml:space="preserve">{2/3} AD x DB &amp;</s>
  <s xml:id="echoid-s13524" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13525" xml:space="preserve">ſegm. </s>
  <s xml:id="echoid-s13526" xml:space="preserve">ADB.</s>
  <s xml:id="echoid-s13527" xml:space="preserve"/>
</p>
<pb o="100" file="0278" n="293" rhead=""/>
<p>
  <s xml:id="echoid-s13528" xml:space="preserve">XXI. </s>
  <s xml:id="echoid-s13529" xml:space="preserve">Porrò, ſit _circuli_ (cujus centrum C) ſegmentum BAE, cu-<lb/>jus axis AD, &amp; </s>
  <s xml:id="echoid-s13530" xml:space="preserve">_gravitatis centrum_ K; </s>
  <s xml:id="echoid-s13531" xml:space="preserve">ponatur autem AD = <lb/>{_s_/_t_} CA, &amp; </s>
  <s xml:id="echoid-s13532" xml:space="preserve">HD = {2 _t_ - _s_/5 _t_ - 3 _s_} AD; </s>
  <s xml:id="echoid-s13533" xml:space="preserve">erit HD major ipsâ KD.</s>
  <s xml:id="echoid-s13534" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13535" xml:space="preserve">Nam per H ducatur recta OP ad BE parallela; </s>
  <s xml:id="echoid-s13536" xml:space="preserve">éſtque punctum <lb/>
<anchor type="note" xlink:label="note-0278-01a" xlink:href="note-0278-01"/>
H <anchor type="note" xlink:href="" symbol="(_a_)"/> centrum _gravitatis paraboliformis_, (puta AF B) ad baſin B E
<anchor type="note" xlink:label="note-0278-02a" xlink:href="note-0278-02"/>
conſtitutæ, cujus exponens {_t_ - _s_/2 _t_ - _s_} &amp; </s>
  <s xml:id="echoid-s13537" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_b_)"/> quæ proinde circulum AEB
<anchor type="note" xlink:label="note-0278-03a" xlink:href="note-0278-03"/>
tangit; </s>
  <s xml:id="echoid-s13538" xml:space="preserve">(nam ſi {_t_ - _s_/2 _t_ - _s_} = {_n_/_m_}; </s>
  <s xml:id="echoid-s13539" xml:space="preserve">erit {2 _t_ - _s_/5 _t_ - 3 _s_} = {_m_/_n_ + 2 _m_}) &amp; </s>
  <s xml:id="echoid-s13540" xml:space="preserve">pro-<lb/>inde H <anchor type="note" xlink:href="" symbol="(_a_)"/> erit centrum gravitatis _paraboliformis_ iſti coordinatæ per
O, P tranſeuntis, &amp; </s>
  <s xml:id="echoid-s13541" xml:space="preserve">ad baſin BE pertingentis. </s>
  <s xml:id="echoid-s13542" xml:space="preserve">Hæc autem ſupra O <lb/>P <anchor type="note" xlink:href="" symbol="(_c_)"/> extra _circulum_ cadit, &amp; </s>
  <s xml:id="echoid-s13543" xml:space="preserve">infra OP <anchor type="note" xlink:href="" symbol="(_d_)"/> intra ipſum; </s>
  <s xml:id="echoid-s13544" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_e_)"/> adeóque
<anchor type="note" xlink:label="note-0278-04a" xlink:href="note-0278-04"/>
<anchor type="note" xlink:label="note-0278-05a" xlink:href="note-0278-05"/>
<anchor type="note" xlink:label="note-0278-06a" xlink:href="note-0278-06"/>
punctum H ſupra K ſitum eſt.</s>
  <s xml:id="echoid-s13545" xml:space="preserve"/>
</p>
<div xml:id="echoid-div419" type="float" level="2" n="21">
<note position="left" xlink:label="note-0278-01" xlink:href="note-0278-01a" xml:space="preserve">Fig. 146.</note>
<note symbol="(_a_)" position="left" xlink:label="note-0278-02" xlink:href="note-0278-02a" xml:space="preserve">2 _hujus ap._</note>
<note symbol="(_b_)" position="left" xlink:label="note-0278-03" xlink:href="note-0278-03a" xml:space="preserve">8. _hujus ap._</note>
<note symbol="(_c_)" position="left" xlink:label="note-0278-04" xlink:href="note-0278-04a" xml:space="preserve">10. _hujus ap_</note>
<note symbol="(_d_)" position="left" xlink:label="note-0278-05" xlink:href="note-0278-05a" xml:space="preserve">11 _hujus ap_</note>
<note symbol="(_e_)" position="left" xlink:label="note-0278-06" xlink:href="note-0278-06a" xml:space="preserve">4. _hujus ap._</note>
</div>
<p>
  <s xml:id="echoid-s13546" xml:space="preserve">XXII. </s>
  <s xml:id="echoid-s13547" xml:space="preserve">Sin punctum L ſit _centrum gravitatis parabolæ_, erit L infra <lb/>
<anchor type="note" xlink:label="note-0278-07a" xlink:href="note-0278-07"/>
K ſitum; </s>
  <s xml:id="echoid-s13548" xml:space="preserve">adeóque KD &amp;</s>
  <s xml:id="echoid-s13549" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13550" xml:space="preserve">{2/5} AD. </s>
  <s xml:id="echoid-s13551" xml:space="preserve">Patet ex 4, &amp; </s>
  <s xml:id="echoid-s13552" xml:space="preserve">17 hujus appen-<lb/>diculæ.</s>
  <s xml:id="echoid-s13553" xml:space="preserve"/>
</p>
<div xml:id="echoid-div420" type="float" level="2" n="22">
<note position="left" xlink:label="note-0278-07" xlink:href="note-0278-07a" xml:space="preserve">Fig. 146.</note>
</div>
<p>
  <s xml:id="echoid-s13554" xml:space="preserve">XXIII. </s>
  <s xml:id="echoid-s13555" xml:space="preserve">Sit _Hyperbolæ_ (cujus centrum C) _ſegmentum_ BAE, cujus <lb/>
<anchor type="note" xlink:label="note-0278-08a" xlink:href="note-0278-08"/>
axis AD, baſis BE; </s>
  <s xml:id="echoid-s13556" xml:space="preserve">gravitatis centrum K; </s>
  <s xml:id="echoid-s13557" xml:space="preserve">ponatur autem AD = <lb/>{_s_ / _t_} CA, &amp; </s>
  <s xml:id="echoid-s13558" xml:space="preserve">HD = {2 _t_ + _s_/5 _t_ + 3 _s_} AD; </s>
  <s xml:id="echoid-s13559" xml:space="preserve">erit HD minor ipsâ<unsure/> KD.</s>
  <s xml:id="echoid-s13560" xml:space="preserve"/>
</p>
<div xml:id="echoid-div421" type="float" level="2" n="23">
<note position="left" xlink:label="note-0278-08" xlink:href="note-0278-08a" xml:space="preserve">Fig. 147.</note>
</div>
<p>
  <s xml:id="echoid-s13561" xml:space="preserve">Nam per H ducatur recta OP ad BE parallela <anchor type="note" xlink:href="" symbol="(_a_)"/>. </s>
  <s xml:id="echoid-s13562" xml:space="preserve">Eſtque punctum
<anchor type="note" xlink:label="note-0278-09a" xlink:href="note-0278-09"/>
H centrum gr. </s>
  <s xml:id="echoid-s13563" xml:space="preserve">_paraboliformis_, puta AFB, ad baſin DB conſtitutæ, <lb/>cujus exponens {_t_ + _s_/2 _t_ + _s_}; </s>
  <s xml:id="echoid-s13564" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_b_)"/> quæ &amp; </s>
  <s xml:id="echoid-s13565" xml:space="preserve">_Hyperbolam_ ad B contingit (nam
<anchor type="note" xlink:label="note-0278-10a" xlink:href="note-0278-10"/>
ſi {_t_ + _s_/2 _t_ + _s_} = {_n_/_m_}; </s>
  <s xml:id="echoid-s13566" xml:space="preserve">erit {2 _t_ + _s_/5 _t_ +3 _s_} = {_m_/_n_ + 2_m_} <anchor type="note" xlink:href="" symbol="(_a_)"/> quare H erit cen-
trum gravitatis paraboliformis iſti coordinatæ per O, P ductæ, &amp; </s>
  <s xml:id="echoid-s13567" xml:space="preserve">ad BE <lb/>pertingentis. </s>
  <s xml:id="echoid-s13568" xml:space="preserve">hæc autem ſupra OP <anchor type="note" xlink:href="" symbol="(_c_)"/> intra hyperbolam cadit;</s>
  <s xml:id="echoid-s13569" xml:space="preserve">
<anchor type="note" xlink:label="note-0278-11a" xlink:href="note-0278-11"/>
&amp; </s>
  <s xml:id="echoid-s13570" xml:space="preserve">infra OP <anchor type="note" xlink:href="" symbol="(_d_)"/> extra illam; </s>
  <s xml:id="echoid-s13571" xml:space="preserve"><anchor type="note" xlink:href="" symbol="(_e_)"/> inde pun@um K ſupra H
<anchor type="note" xlink:label="note-0278-12a" xlink:href="note-0278-12"/>
<anchor type="note" xlink:label="note-0278-13a" xlink:href="note-0278-13"/>
exiſtit.</s>
  <s xml:id="echoid-s13572" xml:space="preserve"/>
</p>
<div xml:id="echoid-div422" type="float" level="2" n="24">
<note symbol="(_a_)" position="left" xlink:label="note-0278-09" xlink:href="note-0278-09a" xml:space="preserve">2. _hujus. ap._</note>
<note symbol="(_b_)" position="left" xlink:label="note-0278-10" xlink:href="note-0278-10a" xml:space="preserve">13. _hujus ap_</note>
<note symbol="(_c_)" position="left" xlink:label="note-0278-11" xlink:href="note-0278-11a" xml:space="preserve">15. _hujus ap._</note>
<note symbol="(_d_)" position="left" xlink:label="note-0278-12" xlink:href="note-0278-12a" xml:space="preserve">16 _hujus ap_</note>
<note symbol="(_e_)" position="left" xlink:label="note-0278-13" xlink:href="note-0278-13a" xml:space="preserve">4. _hujus ap._</note>
</div>
<p>
  <s xml:id="echoid-s13573" xml:space="preserve">XXIV. </s>
  <s xml:id="echoid-s13574" xml:space="preserve">Parabolæ centrum gr. </s>
  <s xml:id="echoid-s13575" xml:space="preserve">(puta L) ſupra K exiſtit, adeóque <lb/>KD &amp;</s>
  <s xml:id="echoid-s13576" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13577" xml:space="preserve">{2/3} AD. </s>
  <s xml:id="echoid-s13578" xml:space="preserve">Patet ex 4, &amp; </s>
  <s xml:id="echoid-s13579" xml:space="preserve">18 hujus appendiculæ.</s>
  <s xml:id="echoid-s13580" xml:space="preserve"/>
</p>
<pb o="101" file="0279" n="294" rhead=""/>
<p>
  <s xml:id="echoid-s13581" xml:space="preserve">XXV. </s>
  <s xml:id="echoid-s13582" xml:space="preserve">Nè ſpeculatio præſens, _ob bujuſmodi complures metbodos Cy-_ <lb/>_clometricas indies promulgatas_, aſpernanda videatur, adjungemus con-<lb/>ſectarium unum vel alterum, quibus fortè ſolis hæc paucula meruerant <lb/>
<anchor type="note" xlink:label="note-0279-01a" xlink:href="note-0279-01"/>
impendi; </s>
  <s xml:id="echoid-s13583" xml:space="preserve">à quibus nempe _Maxima, Minimaque_ ſui generis innume-<lb/>ra determinantur.</s>
  <s xml:id="echoid-s13584" xml:space="preserve"/>
</p>
<div xml:id="echoid-div423" type="float" level="2" n="25">
<note position="right" xlink:label="note-0279-01" xlink:href="note-0279-01a" xml:space="preserve">Fig. 148.</note>
</div>
<p>
  <s xml:id="echoid-s13585" xml:space="preserve">Sit _Semicirculus_ ABZ, cujus centrum C; </s>
  <s xml:id="echoid-s13586" xml:space="preserve">ſitque _ſegmentum<unsure/>_ <lb/>ADB; </s>
  <s xml:id="echoid-s13587" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13588" xml:space="preserve">huic adſcripta _paraboliformis_ AFB, cujus exponens {_u_/_m_}; <lb/></s>
  <s xml:id="echoid-s13589" xml:space="preserve">ſit item AD = {_m_ - 2 _n_/_m_ - _n_} CA; </s>
  <s xml:id="echoid-s13590" xml:space="preserve">_paraboliform@s_ autem _parameter_ (hoc <lb/>eſt recta, cujus aliqua poteſtas in poteſtatem ſegmenti axis, ſeu AD, <lb/>ducta conficit _poteſtatem_ ordinatæ, ceu D B) nominetur _p_; </s>
  <s xml:id="echoid-s13591" xml:space="preserve">erit _p_ in ſuo <lb/>genere _maximum_.</s>
  <s xml:id="echoid-s13592" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13593" xml:space="preserve">Nam utcunque ducatur GE ad DB parallela, &amp; </s>
  <s xml:id="echoid-s13594" xml:space="preserve">ad GE poſita <lb/>concipiatur _paraboliformis_, ipſi AFB coordinata, cujus _parameter_ di-<lb/>catur _q_. </s>
  <s xml:id="echoid-s13595" xml:space="preserve">quum ergò _paraboliformis_ AFB _circulum_ extrorſum contin-<lb/>gat, erit GF &amp;</s>
  <s xml:id="echoid-s13596" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13597" xml:space="preserve">GE; </s>
  <s xml:id="echoid-s13598" xml:space="preserve">adeóque GF {_m_/ } &amp;</s>
  <s xml:id="echoid-s13599" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13600" xml:space="preserve">GE {_m_/ }; </s>
  <s xml:id="echoid-s13601" xml:space="preserve">hoc eſt _p_ {_m_ - _n_/ } x <lb/>AG {_n_/ } &amp;</s>
  <s xml:id="echoid-s13602" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13603" xml:space="preserve">q {_m_ - _n_/ } x AG {_n_/ }; </s>
  <s xml:id="echoid-s13604" xml:space="preserve">quare _p_ &amp;</s>
  <s xml:id="echoid-s13605" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13606" xml:space="preserve">_q_.</s>
  <s xml:id="echoid-s13607" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13608" xml:space="preserve">Notandum eſt eſſe _p_ {2 _m_ - 2 _n_/ } = ZD<emph style="sub">_m_</emph> x AD {_m_ - 2 _n_/ }. </s>
  <s xml:id="echoid-s13609" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13610" xml:space="preserve">q {2 _m_ - 2 _n_/ } <lb/>= ZG {_m_/ } x AG {_m_ - 2 _n_/ }. </s>
  <s xml:id="echoid-s13611" xml:space="preserve">unde ZD {_m_/ } x AD {_m_ - 2 _n_/ } &amp;</s>
  <s xml:id="echoid-s13612" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13613" xml:space="preserve">ZG {_m_/ } x AG {_m_ - 2 _n_/ }. <lb/></s>
  <s xml:id="echoid-s13614" xml:space="preserve">quare ZD {_m_/ } x AD {_m_ - 2 _n_/ } eſt maximum.</s>
  <s xml:id="echoid-s13615" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13616" xml:space="preserve">Exemp. </s>
  <s xml:id="echoid-s13617" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13618" xml:space="preserve">Sit _n_ = 1, &amp; </s>
  <s xml:id="echoid-s13619" xml:space="preserve">_m_ = 3. </s>
  <s xml:id="echoid-s13620" xml:space="preserve">erit ideò _p_ {4/ } = ZD {3/ } x AD = <lb/>ZD_q_ x BD_q_; </s>
  <s xml:id="echoid-s13621" xml:space="preserve">vel _p_<emph style="sub">2</emph> = ZD x BD. </s>
  <s xml:id="echoid-s13622" xml:space="preserve">Item AD = <lb/>{1/2} CA.</s>
  <s xml:id="echoid-s13623" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13624" xml:space="preserve">2. </s>
  <s xml:id="echoid-s13625" xml:space="preserve">Sit _n_ = 3, &amp; </s>
  <s xml:id="echoid-s13626" xml:space="preserve">_m_ = 10. </s>
  <s xml:id="echoid-s13627" xml:space="preserve">erit p {14/ } = ZD<emph style="sub">10</emph> x AD<emph style="sub">4</emph>. <lb/></s>
  <s xml:id="echoid-s13628" xml:space="preserve">vel p {7/ } = ZD {5/ } x AD<emph style="sub">2</emph> = ZD<emph style="sub">3</emph> x BD<emph style="sub">4</emph>. </s>
  <s xml:id="echoid-s13629" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13630" xml:space="preserve">AD <lb/>= {4/7} CA.</s>
  <s xml:id="echoid-s13631" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13632" xml:space="preserve">XXVI. </s>
  <s xml:id="echoid-s13633" xml:space="preserve">Sit item _hyperbola_ (æquilatera) cujus centrum C, axis <lb/>ZA; </s>
  <s xml:id="echoid-s13634" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13635" xml:space="preserve">huic inſcripta _paraboliformis_ AFB cujus expo-<lb/>
<anchor type="note" xlink:label="note-0279-02a" xlink:href="note-0279-02"/>
nens {_n_/_m_} _parameter p_; </s>
  <s xml:id="echoid-s13636" xml:space="preserve">ſitque AD = {2_n_ - _m_/_m_ - _n_} CA; </s>
  <s xml:id="echoid-s13637" xml:space="preserve">erit _p_ ſui gene-<lb/>neris maximum.</s>
  <s xml:id="echoid-s13638" xml:space="preserve"/>
</p>
<div xml:id="echoid-div424" type="float" level="2" n="26">
<note position="right" xlink:label="note-0279-02" xlink:href="note-0279-02a" xml:space="preserve">Fig. 149.</note>
</div>
<p>
  <s xml:id="echoid-s13639" xml:space="preserve">Nam utcunque ducatur EG ad BD parallela; </s>
  <s xml:id="echoid-s13640" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13641" xml:space="preserve">ad EG conſtituta <lb/>intelligatur _paraboliformis_, ipſi AFB coordinata, cujus _parameter q._ <lb/></s>
  <s xml:id="echoid-s13642" xml:space="preserve">quum ergo _paraboliformis_ AFB _hyperbolam_ introrſum contingat, <lb/>erit GF {_m_/ } &amp;</s>
  <s xml:id="echoid-s13643" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13644" xml:space="preserve">GE {_n_/ }; </s>
  <s xml:id="echoid-s13645" xml:space="preserve">hoc eſt _p_ {_m_ - _n_/ } x AG<emph style="sub">n</emph> &amp;</s>
  <s xml:id="echoid-s13646" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13647" xml:space="preserve">_q_ {_m_ - _n_/ } x AG<emph style="sub">n</emph>; </s>
  <s xml:id="echoid-s13648" xml:space="preserve"><lb/>quare _p_ &amp;</s>
  <s xml:id="echoid-s13649" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13650" xml:space="preserve">_q_.</s>
  <s xml:id="echoid-s13651" xml:space="preserve"/>
</p>
<pb o="102" file="0280" n="295" rhead=""/>
<p>
  <s xml:id="echoid-s13652" xml:space="preserve">_Notandum_ eſt eſſe _p_ {2 _m_ - 2 _n_/ } = {ZD {_m_/ }/AD {2 _n_ - _m_/ }}. </s>
  <s xml:id="echoid-s13653" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13654" xml:space="preserve">_q_ {2 _m_ - 2 _n_/ } = <lb/>{ZG {_m_/ }/AG {2 _n_ - _m_/ }}. </s>
  <s xml:id="echoid-s13655" xml:space="preserve">unde erit {ZD {_m_/ }/AD {2 _n_ - _m_/ }} &amp;</s>
  <s xml:id="echoid-s13656" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s13657" xml:space="preserve">{ZG {_m_/ }/AG {2 _n_ - _m_/ }}. </s>
  <s xml:id="echoid-s13658" xml:space="preserve">quare {ZD {_m_/ }/AD {2 _n_ - _m_/ }} eſt mi-<lb/>
<anchor type="note" xlink:label="note-0280-01a" xlink:href="note-0280-01"/>
nimum.</s>
  <s xml:id="echoid-s13659" xml:space="preserve"/>
</p>
<div xml:id="echoid-div425" type="float" level="2" n="27">
<note position="left" xlink:label="note-0280-01" xlink:href="note-0280-01a" xml:space="preserve">Fig. 149.</note>
</div>
<p>
  <s xml:id="echoid-s13660" xml:space="preserve">_Exemp_. </s>
  <s xml:id="echoid-s13661" xml:space="preserve">1. </s>
  <s xml:id="echoid-s13662" xml:space="preserve">Sit _n_ = 2; </s>
  <s xml:id="echoid-s13663" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13664" xml:space="preserve">_m_ = 3; </s>
  <s xml:id="echoid-s13665" xml:space="preserve">erit _p_<emph style="sub">2</emph> = {ZD<emph style="sub">3</emph>/AD}. </s>
  <s xml:id="echoid-s13666" xml:space="preserve">= {BD<emph style="sub">6</emph>/AD<emph style="sub">4</emph>}. </s>
  <s xml:id="echoid-s13667" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13668" xml:space="preserve"><lb/>_p_ = {BD<emph style="sub">3</emph>/ADq}. </s>
  <s xml:id="echoid-s13669" xml:space="preserve">= {ZDq/BD}. </s>
  <s xml:id="echoid-s13670" xml:space="preserve">Item AD = CA.</s>
  <s xml:id="echoid-s13671" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13672" xml:space="preserve">2. </s>
  <s xml:id="echoid-s13673" xml:space="preserve">Sit _n_ = 3; </s>
  <s xml:id="echoid-s13674" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13675" xml:space="preserve">_m_ = 4; </s>
  <s xml:id="echoid-s13676" xml:space="preserve">erit _p_<emph style="sub">2</emph> = {ZD<emph style="sub">4</emph>/AD<emph style="sub">2</emph>} vel p = {ZD<emph style="sub">2</emph>/AD} <lb/> = {BD<emph style="sub">4</emph>/AD<emph style="sub">3</emph>} = {ZD<emph style="sub">3</emph>/BD<emph style="sub">2</emph>}. </s>
  <s xml:id="echoid-s13677" xml:space="preserve">Item AD = 2 CA.</s>
  <s xml:id="echoid-s13678" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13679" xml:space="preserve">3. </s>
  <s xml:id="echoid-s13680" xml:space="preserve">Sit _n_ = 5, &amp; </s>
  <s xml:id="echoid-s13681" xml:space="preserve">_m_ = 8; </s>
  <s xml:id="echoid-s13682" xml:space="preserve">erit _p_ {6/ } = {ZD<emph style="sub">8</emph>/AD<emph style="sub">2</emph>}. </s>
  <s xml:id="echoid-s13683" xml:space="preserve">vel _p_<emph style="sub">3</emph> = {ZD<emph style="sub">4</emph>/AD} <lb/> = {BD<emph style="sub">8</emph>/AD<emph style="sub">5</emph>} = {ZD<emph style="sub">5</emph>/BD<emph style="sub">2</emph>}. </s>
  <s xml:id="echoid-s13684" xml:space="preserve">Item AD = {2/3} CA.</s>
  <s xml:id="echoid-s13685" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13686" xml:space="preserve">Quoniam in his _Cyclometriam_ attigi, quid ſi obiter eò ſpectantia <lb/>_Theoremata_, quæ ad manum, paucula ſubjunxero? </s>
  <s xml:id="echoid-s13687" xml:space="preserve">præſternatur au-<lb/>
<anchor type="note" xlink:label="note-0280-02a" xlink:href="note-0280-02"/>
tem autem hoc χαυολικὸν:</s>
  <s xml:id="echoid-s13688" xml:space="preserve"/>
</p>
<div xml:id="echoid-div426" type="float" level="2" n="28">
<note position="left" xlink:label="note-0280-02" xlink:href="note-0280-02a" xml:space="preserve">Fig. 150.</note>
</div>
<p>
  <s xml:id="echoid-s13689" xml:space="preserve">Sit curva quæpiam AGB, cujus axis AD, &amp; </s>
  <s xml:id="echoid-s13690" xml:space="preserve">ad hunc ordinatæ <lb/>rectæ BD, GE; </s>
  <s xml:id="echoid-s13691" xml:space="preserve">Habebit curva AB ad curvam AG majorem rati-<lb/>onem quàm recta BD ad rectam GE.</s>
  <s xml:id="echoid-s13692" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13693" xml:space="preserve">Nam ducatur recta GH ad AD parallela: </s>
  <s xml:id="echoid-s13694" xml:space="preserve">ſecentúrque recta B H <lb/>punctis Y, &amp; </s>
  <s xml:id="echoid-s13695" xml:space="preserve">recta GE punctis Z in particulas indefinitè multas; </s>
  <s xml:id="echoid-s13696" xml:space="preserve">pér-<lb/>que puncta Y, Z ducantur rectæ YM, YN, ZO, ZP ad AD paral-<lb/>lelæ: </s>
  <s xml:id="echoid-s13697" xml:space="preserve">curvam interſecantes punctis M, N, O, P; </s>
  <s xml:id="echoid-s13698" xml:space="preserve">per quæ ducantur <lb/>rectæ MR, NS, OT, PV ad BD parallelæ. </s>
  <s xml:id="echoid-s13699" xml:space="preserve">Eſtque angulus BM Y <lb/>(ut è ſuperius oſtenſis liquet) minor angulo NGS, unde MB. </s>
  <s xml:id="echoid-s13700" xml:space="preserve">B Y <lb/>&amp;</s>
  <s xml:id="echoid-s13701" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13702" xml:space="preserve">GN. </s>
  <s xml:id="echoid-s13703" xml:space="preserve">NS. </s>
  <s xml:id="echoid-s13704" xml:space="preserve">Similíque de cauſa eſt NM. </s>
  <s xml:id="echoid-s13705" xml:space="preserve">MR &amp;</s>
  <s xml:id="echoid-s13706" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13707" xml:space="preserve">GN. </s>
  <s xml:id="echoid-s13708" xml:space="preserve">NS. </s>
  <s xml:id="echoid-s13709" xml:space="preserve"><anchor type="note" xlink:href="" symbol="*"/>
<anchor type="note" xlink:label="note-0280-03a" xlink:href="note-0280-03"/>
quare conjunctè eft BM + MN + NG. </s>
  <s xml:id="echoid-s13710" xml:space="preserve">BY + MR + NS &amp;</s>
  <s xml:id="echoid-s13711" xml:space="preserve">gt; <lb/></s>
  <s xml:id="echoid-s13712" xml:space="preserve">GN. </s>
  <s xml:id="echoid-s13713" xml:space="preserve">NS; </s>
  <s xml:id="echoid-s13714" xml:space="preserve">hoc eſt arc. </s>
  <s xml:id="echoid-s13715" xml:space="preserve">GB. </s>
  <s xml:id="echoid-s13716" xml:space="preserve">BH &amp;</s>
  <s xml:id="echoid-s13717" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13718" xml:space="preserve">GN. </s>
  <s xml:id="echoid-s13719" xml:space="preserve">NS. </s>
  <s xml:id="echoid-s13720" xml:space="preserve">rurſus (è diſcur-<lb/>ſu conſimili) ratio GN ad NS major eſt ſingulis rationibus OG ad <lb/>GZ, OP ad PT, &amp; </s>
  <s xml:id="echoid-s13721" xml:space="preserve">AP ad PV; </s>
  <s xml:id="echoid-s13722" xml:space="preserve">idcircoq; </s>
  <s xml:id="echoid-s13723" xml:space="preserve">junctè eſt GN. </s>
  <s xml:id="echoid-s13724" xml:space="preserve">NS &amp;</s>
  <s xml:id="echoid-s13725" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13726" xml:space="preserve"><lb/>arc. </s>
  <s xml:id="echoid-s13727" xml:space="preserve">AG. </s>
  <s xml:id="echoid-s13728" xml:space="preserve">GE. </s>
  <s xml:id="echoid-s13729" xml:space="preserve">quapropter erit GB. </s>
  <s xml:id="echoid-s13730" xml:space="preserve">BH &amp;</s>
  <s xml:id="echoid-s13731" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13732" xml:space="preserve">AG. </s>
  <s xml:id="echoid-s13733" xml:space="preserve">GE. </s>
  <s xml:id="echoid-s13734" xml:space="preserve">permutan-<lb/>doque GB. </s>
  <s xml:id="echoid-s13735" xml:space="preserve">AG &amp;</s>
  <s xml:id="echoid-s13736" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13737" xml:space="preserve">BH. </s>
  <s xml:id="echoid-s13738" xml:space="preserve">GE. </s>
  <s xml:id="echoid-s13739" xml:space="preserve">quare componendo eſt AB. </s>
  <s xml:id="echoid-s13740" xml:space="preserve">A G <lb/>&amp;</s>
  <s xml:id="echoid-s13741" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13742" xml:space="preserve">BD. </s>
  <s xml:id="echoid-s13743" xml:space="preserve">GE.</s>
  <s xml:id="echoid-s13744" xml:space="preserve"/>
</p>
<div xml:id="echoid-div427" type="float" level="2" n="29">
<note symbol="*" position="left" xlink:label="note-0280-03" xlink:href="note-0280-03a" xml:space="preserve">Vid. _Append_. <lb/>Lect. XII.</note>
</div>
<pb o="103" file="0281" n="296" rhead=""/>
<p>
  <s xml:id="echoid-s13745" xml:space="preserve">XXVIII. </s>
  <s xml:id="echoid-s13746" xml:space="preserve">Sit _Circulus_ AMB, cujus _Radiui_ CA, &amp; </s>
  <s xml:id="echoid-s13747" xml:space="preserve">ad hunc per-<lb/>
<anchor type="note" xlink:label="note-0281-01a" xlink:href="note-0281-01"/>
pendicularis recta DBE; </s>
  <s xml:id="echoid-s13748" xml:space="preserve">ſit item curva ANE talis, ut ductâ utcun-<lb/>que rectà PMN ad DE parallelâ (quæ circulum ſecet in M, dictam <lb/>curvam in N) ſit recta PN æqualis _Arcui_ AM; </s>
  <s xml:id="echoid-s13749" xml:space="preserve">ſit demum _axe_ <lb/>AD _baſe_ DE deſcripta _Parabola_ AOE, hæc extra curvam AN E <lb/>tota cadet.</s>
  <s xml:id="echoid-s13750" xml:space="preserve"/>
</p>
<div xml:id="echoid-div428" type="float" level="2" n="30">
<note position="right" xlink:label="note-0281-01" xlink:href="note-0281-01a" xml:space="preserve">Fig. 151</note>
</div>
<p>
  <s xml:id="echoid-s13751" xml:space="preserve">Nam ſecet recta PN parabolam in O; </s>
  <s xml:id="echoid-s13752" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13753" xml:space="preserve">connectantur ſubtenſæ <lb/>AB, AM; </s>
  <s xml:id="echoid-s13754" xml:space="preserve">eſtque DE. </s>
  <s xml:id="echoid-s13755" xml:space="preserve">PN:</s>
  <s xml:id="echoid-s13756" xml:space="preserve">: arc AB. </s>
  <s xml:id="echoid-s13757" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s13758" xml:space="preserve">AM &amp;</s>
  <s xml:id="echoid-s13759" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s13760" xml:space="preserve">AB. </s>
  <s xml:id="echoid-s13761" xml:space="preserve">AM <lb/>:</s>
  <s xml:id="echoid-s13762" xml:space="preserve">: DE. </s>
  <s xml:id="echoid-s13763" xml:space="preserve">PO. </s>
  <s xml:id="echoid-s13764" xml:space="preserve">quare PN&amp;</s>
  <s xml:id="echoid-s13765" xml:space="preserve">lt;</s>
  <s xml:id="echoid-s13766" xml:space="preserve">PO; </s>
  <s xml:id="echoid-s13767" xml:space="preserve">unde liquet Propoſitum.</s>
  <s xml:id="echoid-s13768" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13769" xml:space="preserve">XXIX. </s>
  <s xml:id="echoid-s13770" xml:space="preserve">Exhinc (&amp; </s>
  <s xml:id="echoid-s13771" xml:space="preserve">è vulgò notis _ſpatiorune_ ADB, ADE _dimen-_ <lb/>_ſionibus_) facilè colligitur hæc regula:</s>
  <s xml:id="echoid-s13772" xml:space="preserve">{3 CAx DB/2 CA+CD} &amp;</s>
  <s xml:id="echoid-s13773" xml:space="preserve">lt;</s>
  <s xml:id="echoid-s13774" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s13775" xml:space="preserve">AB.</s>
  <s xml:id="echoid-s13776" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 152.</note>
<p>
  <s xml:id="echoid-s13777" xml:space="preserve">Porrò ſi ponatur arc. </s>
  <s xml:id="echoid-s13778" xml:space="preserve">AB = 30 grad. </s>
  <s xml:id="echoid-s13779" xml:space="preserve">ſitque 2 CA = 113; </s>
  <s xml:id="echoid-s13780" xml:space="preserve">juxta <lb/>regulam iſtam computando, proveniet _tota circumferentia_ major quàm <lb/>355, minus fractione unitatis.</s>
  <s xml:id="echoid-s13781" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13782" xml:space="preserve">XXX. </s>
  <s xml:id="echoid-s13783" xml:space="preserve">Hinc etiam _dato arcu_ AB, nominatiſque AB = p; </s>
  <s xml:id="echoid-s13784" xml:space="preserve">CA = r; </s>
  <s xml:id="echoid-s13785" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13786" xml:space="preserve"><lb/>DB = _e_, ad inveniendum _ſinum rectum_ DB adhibebitur hæc æqua-<lb/>tio; </s>
  <s xml:id="echoid-s13787" xml:space="preserve">{3 _rrpp_/_9rr_ + _pp_} = {12 _rrp_/9 _rr_ + _pp_} _e_-_ee._ </s>
  <s xml:id="echoid-s13788" xml:space="preserve">vel ponendo _k_ = {3 _rrp_/9 _rr_ + _pp_}; </s>
  <s xml:id="echoid-s13789" xml:space="preserve">erit <lb/>_kp_ = 4_ke_ - _ee._ </s>
  <s xml:id="echoid-s13790" xml:space="preserve">vel 2 _k_ - √ 4 _kk_ - _kp_ = _e._</s>
  <s xml:id="echoid-s13791" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13792" xml:space="preserve">XXXI. </s>
  <s xml:id="echoid-s13793" xml:space="preserve">Sit AMB _Circulus_, cujus Radius CA, &amp; </s>
  <s xml:id="echoid-s13794" xml:space="preserve">huic perpendi-<lb/>
<anchor type="note" xlink:label="note-0281-03a" xlink:href="note-0281-03"/>
cularis recta DBE; </s>
  <s xml:id="echoid-s13795" xml:space="preserve">ſit item curva ANE pars _Cycloidis_ ad _Circulum_ <lb/>AMB pertinentis; </s>
  <s xml:id="echoid-s13796" xml:space="preserve">demum ad axem AD, baſin DE ſtatuatur _Para-_ <lb/>_bola_ AOE; </s>
  <s xml:id="echoid-s13797" xml:space="preserve">hæc intra _Cycloidem_ tota cadet.</s>
  <s xml:id="echoid-s13798" xml:space="preserve"/>
</p>
<div xml:id="echoid-div429" type="float" level="2" n="31">
<note position="right" xlink:label="note-0281-03" xlink:href="note-0281-03a" xml:space="preserve">Fig. 153.</note>
</div>
<p>
  <s xml:id="echoid-s13799" xml:space="preserve">Etenim utcunque ducatur recta PM ON ad DE parallela, lineas <lb/>expoſitas ſecans, ut cernis; </s>
  <s xml:id="echoid-s13800" xml:space="preserve">connectantúrque _ſabtenſæ_ AB, AM; <lb/></s>
  <s xml:id="echoid-s13801" xml:space="preserve">eſtque DE. </s>
  <s xml:id="echoid-s13802" xml:space="preserve">PO:</s>
  <s xml:id="echoid-s13803" xml:space="preserve">: AB. </s>
  <s xml:id="echoid-s13804" xml:space="preserve">AM :</s>
  <s xml:id="echoid-s13805" xml:space="preserve">: curv. </s>
  <s xml:id="echoid-s13806" xml:space="preserve">AE. </s>
  <s xml:id="echoid-s13807" xml:space="preserve">AN &amp;</s>
  <s xml:id="echoid-s13808" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13809" xml:space="preserve">DE. </s>
  <s xml:id="echoid-s13810" xml:space="preserve">PN; </s>
  <s xml:id="echoid-s13811" xml:space="preserve"><lb/>adeoque PO &amp;</s>
  <s xml:id="echoid-s13812" xml:space="preserve">lt;</s>
  <s xml:id="echoid-s13813" xml:space="preserve">PN. </s>
  <s xml:id="echoid-s13814" xml:space="preserve">unde conſtat Propoſitum.</s>
  <s xml:id="echoid-s13815" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13816" xml:space="preserve">XXXII. </s>
  <s xml:id="echoid-s13817" xml:space="preserve">Exhinc, &amp; </s>
  <s xml:id="echoid-s13818" xml:space="preserve">è _notis ſegmentorum circular is atque Cycloida-_ <lb/>_lis dimenſionibus_, hæc elicitur _Regula_ {2CA x DB + CD x DB/CA + 2CD} <lb/>&amp;</s>
  <s xml:id="echoid-s13819" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s13820" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s13821" xml:space="preserve">AB.</s>
  <s xml:id="echoid-s13822" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13823" xml:space="preserve">Porrò ſi fuerit arc. </s>
  <s xml:id="echoid-s13824" xml:space="preserve">AB = 30 grad. </s>
  <s xml:id="echoid-s13825" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13826" xml:space="preserve">ponatur 2 CA = 113; </s>
  <s xml:id="echoid-s13827" xml:space="preserve">è <lb/>regula hac conſectatur fore _totam circumferentiam_ minorem quam <lb/>355, plus fractione.</s>
  <s xml:id="echoid-s13828" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13829" xml:space="preserve">Vides igitur ut è propoſitis duabus regulis ſtatim emergit _Diametri_ <lb/>ad _Circumferentiam Proportio Metiana_.</s>
  <s xml:id="echoid-s13830" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13831" xml:space="preserve">XXXIII. </s>
  <s xml:id="echoid-s13832" xml:space="preserve">Quoniam exorbitanti ſe obviam dedit _Cyclois_ hoc adno-<lb/>tabo _@ beorema_, neſcio an uſpiam ab illis, qui de _Cycloide_ tam fusè <lb/>ſcripſerunt, animadverſum; </s>
  <s xml:id="echoid-s13833" xml:space="preserve">Completo _Rectangulo_ ADEG, _ſpatium_
<pb o="104" file="0282" n="297" rhead=""/>
AEG æquatur _Circulari ſegmento_ ADB demonſtrationem, ne longiùs <lb/>evager, obmittam.</s>
  <s xml:id="echoid-s13834" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13835" xml:space="preserve">XXXIV. </s>
  <s xml:id="echoid-s13836" xml:space="preserve">Sint duo _circuli_ AIMG, AKNH ſeſe contingentes ad A; <lb/></s>
  <s xml:id="echoid-s13837" xml:space="preserve">
<anchor type="note" xlink:label="note-0282-01a" xlink:href="note-0282-01"/>
communique diametro AHG, utcunque perpendicularis ducatur recta <lb/>DN M: </s>
  <s xml:id="echoid-s13838" xml:space="preserve">habebit_ſegmentum_ AIMD ad _ſegmentum_ AKND mino-<lb/>rem rationem, quam recta DM ad rectam DN.</s>
  <s xml:id="echoid-s13839" xml:space="preserve"/>
</p>
<div xml:id="echoid-div430" type="float" level="2" n="32">
<note position="left" xlink:label="note-0282-01" xlink:href="note-0282-01a" xml:space="preserve">Fig. 154.</note>
</div>
<p>
  <s xml:id="echoid-s13840" xml:space="preserve">Nam ſit AR ad AG perpendicularis, ac ipſi AH æqualis; </s>
  <s xml:id="echoid-s13841" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13842" xml:space="preserve"><lb/>connectatur HR, cui occurrat recta MD in X; </s>
  <s xml:id="echoid-s13843" xml:space="preserve">ducatúrque recta <lb/>GXS; </s>
  <s xml:id="echoid-s13844" xml:space="preserve">tum ad axem AG _parametrum_ AS per N deſcripta con-<lb/>cipiatur _Ellipſis_ ALNG; </s>
  <s xml:id="echoid-s13845" xml:space="preserve">hæc (utì ſatis manifeſtum) intra arcum <lb/>AKN tota cadet. </s>
  <s xml:id="echoid-s13846" xml:space="preserve">Eſt autem ſegm. </s>
  <s xml:id="echoid-s13847" xml:space="preserve">AIMD. </s>
  <s xml:id="echoid-s13848" xml:space="preserve">ſegm. </s>
  <s xml:id="echoid-s13849" xml:space="preserve">ALND:</s>
  <s xml:id="echoid-s13850" xml:space="preserve">: <lb/>DM. </s>
  <s xml:id="echoid-s13851" xml:space="preserve">DN. </s>
  <s xml:id="echoid-s13852" xml:space="preserve">ergo ſegm. </s>
  <s xml:id="echoid-s13853" xml:space="preserve">AI MD. </s>
  <s xml:id="echoid-s13854" xml:space="preserve">ſegm. </s>
  <s xml:id="echoid-s13855" xml:space="preserve">AKND &amp;</s>
  <s xml:id="echoid-s13856" xml:space="preserve">lt;</s>
  <s xml:id="echoid-s13857" xml:space="preserve">DM. </s>
  <s xml:id="echoid-s13858" xml:space="preserve">DN.</s>
  <s xml:id="echoid-s13859" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13860" xml:space="preserve">XXXV. </s>
  <s xml:id="echoid-s13861" xml:space="preserve">Sit Ellipſis YFZT, cujus axes conjugati YZ, FT; </s>
  <s xml:id="echoid-s13862" xml:space="preserve">ſit item <lb/>recta DC axi majori YZ parallela; </s>
  <s xml:id="echoid-s13863" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13864" xml:space="preserve">per D, F, C tranſeat circulus <lb/>
<anchor type="note" xlink:label="note-0282-02a" xlink:href="note-0282-02"/>
DFCV centrum habens K, in ellipſis axe minore FT ſitum; </s>
  <s xml:id="echoid-s13865" xml:space="preserve">dico <lb/>circuli partem DOFPC intra ellipſis partem DMFNC jacere.</s>
  <s xml:id="echoid-s13866" xml:space="preserve"/>
</p>
<div xml:id="echoid-div431" type="float" level="2" n="33">
<note position="left" xlink:label="note-0282-02" xlink:href="note-0282-02a" xml:space="preserve">Fig + 154.</note>
</div>
<p>
  <s xml:id="echoid-s13867" xml:space="preserve">Nam ſit FI ad FV perpendicularis, &amp; </s>
  <s xml:id="echoid-s13868" xml:space="preserve">in hac ſumatur FS = FV; </s>
  <s xml:id="echoid-s13869" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13870" xml:space="preserve"><lb/>connectatur VS, cui DC producta occurrat in X; </s>
  <s xml:id="echoid-s13871" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13872" xml:space="preserve">connexa TX <lb/>ipſi FI occurat in R. </s>
  <s xml:id="echoid-s13873" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13874" xml:space="preserve">cum ſit GDq = FG x GV = FG x GX; <lb/></s>
  <s xml:id="echoid-s13875" xml:space="preserve">liquet ipſam FR eſſe ellipſis, axi FT congruam, parametrum; </s>
  <s xml:id="echoid-s13876" xml:space="preserve">unde <lb/>conſtat Propoſitum.</s>
  <s xml:id="echoid-s13877" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13878" xml:space="preserve">XXXVI. </s>
  <s xml:id="echoid-s13879" xml:space="preserve">Sit circuli, cujus centrum L, ſegmentum DEC, &amp; </s>
  <s xml:id="echoid-s13880" xml:space="preserve">ſumpto <lb/>
<anchor type="note" xlink:label="note-0282-03a" xlink:href="note-0282-03"/>
in ejus axe GE puncto quopiam F, ſit curva DMFC talis, ut ductâ <lb/>utcunque rectâ RMS ad GE parallelâ, ſit RS. </s>
  <s xml:id="echoid-s13881" xml:space="preserve">RM:</s>
  <s xml:id="echoid-s13882" xml:space="preserve">: GE. </s>
  <s xml:id="echoid-s13883" xml:space="preserve">GF; <lb/></s>
  <s xml:id="echoid-s13884" xml:space="preserve">erit DMFC ellipſis, hoc modo determinata: </s>
  <s xml:id="echoid-s13885" xml:space="preserve">Fiat EG. </s>
  <s xml:id="echoid-s13886" xml:space="preserve">FG:</s>
  <s xml:id="echoid-s13887" xml:space="preserve">: GL. </s>
  <s xml:id="echoid-s13888" xml:space="preserve"><lb/>GH; </s>
  <s xml:id="echoid-s13889" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13890" xml:space="preserve">per H erigatur YHZ ad DC parallela, ſitque HY par ipſi LE; </s>
  <s xml:id="echoid-s13891" xml:space="preserve"><lb/>erunt HY, HF ellipſis ſemiaxes.</s>
  <s xml:id="echoid-s13892" xml:space="preserve"/>
</p>
<div xml:id="echoid-div432" type="float" level="2" n="34">
<note position="left" xlink:label="note-0282-03" xlink:href="note-0282-03a" xml:space="preserve">Fig 155.</note>
</div>
<p>
  <s xml:id="echoid-s13893" xml:space="preserve">Demonſtratum habetur à _Greg. </s>
  <s xml:id="echoid-s13894" xml:space="preserve">à S. </s>
  <s xml:id="echoid-s13895" xml:space="preserve">Vincentio_, L. </s>
  <s xml:id="echoid-s13896" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s13897" xml:space="preserve">Prop. </s>
  <s xml:id="echoid-s13898" xml:space="preserve">154. <lb/></s>
  <s xml:id="echoid-s13899" xml:space="preserve">_Corol._ </s>
  <s xml:id="echoid-s13900" xml:space="preserve">Hinc ſegm. </s>
  <s xml:id="echoid-s13901" xml:space="preserve">DEC. </s>
  <s xml:id="echoid-s13902" xml:space="preserve">DMFC:</s>
  <s xml:id="echoid-s13903" xml:space="preserve">: EG. </s>
  <s xml:id="echoid-s13904" xml:space="preserve">FG.</s>
  <s xml:id="echoid-s13905" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13906" xml:space="preserve">XXXVII. </s>
  <s xml:id="echoid-s13907" xml:space="preserve">Sint duæ circulorum portiones DEC, DOFC, quarum <lb/>communis ſubtenſa DC, &amp; </s>
  <s xml:id="echoid-s13908" xml:space="preserve">axis GFE; </s>
  <s xml:id="echoid-s13909" xml:space="preserve">portio major DEC ad portio-<lb/>nem DOFC majorem rationem habet eâ, quam habet axis GE ad <lb/>axem GF.</s>
  <s xml:id="echoid-s13910" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13911" xml:space="preserve">Nam ſint L circuli DSEC, &amp; </s>
  <s xml:id="echoid-s13912" xml:space="preserve">K circuli DOFC centra; </s>
  <s xml:id="echoid-s13913" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13914" xml:space="preserve">fiat EG. <lb/></s>
  <s xml:id="echoid-s13915" xml:space="preserve">FG:</s>
  <s xml:id="echoid-s13916" xml:space="preserve">: GL. </s>
  <s xml:id="echoid-s13917" xml:space="preserve">GH; </s>
  <s xml:id="echoid-s13918" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13919" xml:space="preserve">fiat YHZ ad HF perpendicularis &amp; </s>
  <s xml:id="echoid-s13920" xml:space="preserve">ſit HY æ-<lb/>qualis ipſi LE; </s>
  <s xml:id="echoid-s13921" xml:space="preserve">tum ſemiaxibus HY, HF deſcripta concipiatur ellipſis <lb/>YDMFCZ; </s>
  <s xml:id="echoid-s13922" xml:space="preserve">è mox prædictis liquet ellipſin DMFC circulo DOFC <lb/>circumduci. </s>
  <s xml:id="echoid-s13923" xml:space="preserve">Eſt autem circulare ſegmentum DEC ad ſegmentum el-<lb/>lipticum DMFC, ut GE ad GF; </s>
  <s xml:id="echoid-s13924" xml:space="preserve">quare ſegm DEC ad ſegm circula-<lb/>re DOFC. </s>
  <s xml:id="echoid-s13925" xml:space="preserve">rationem habet majorem, quàm GE ad GF: </s>
  <s xml:id="echoid-s13926" xml:space="preserve">Quod. </s>
  <s xml:id="echoid-s13927" xml:space="preserve">E.</s>
  <s xml:id="echoid-s13928" xml:space="preserve">D.</s>
  <s xml:id="echoid-s13929" xml:space="preserve"/>
</p>
<pb o="105" file="0283" n="298"/>
</div>
<div xml:id="echoid-div434" type="section" level="1" n="43">
<head xml:id="echoid-head46" xml:space="preserve"><emph style="sc">Lect</emph>. XII.</head>
<p>
  <s xml:id="echoid-s13930" xml:space="preserve">IN ſuſcepto negotio progredimur; </s>
  <s xml:id="echoid-s13931" xml:space="preserve">quod ut (quatenus licet) decurte-<lb/>
<anchor type="note" xlink:label="note-0283-01a" xlink:href="note-0283-01"/>
mus, verbíſque parcamus; </s>
  <s xml:id="echoid-s13932" xml:space="preserve">obſervetur, in ſequentibus ubique _line-_ <lb/>_am_ AB _curvam_ eſſe (quales tractamus) quampiam; </s>
  <s xml:id="echoid-s13933" xml:space="preserve">cujus _Axis_ AD; <lb/></s>
  <s xml:id="echoid-s13934" xml:space="preserve">huic applicatas omnes rectas BD, CA, MF, NG perpendiculares; </s>
  <s xml:id="echoid-s13935" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s13936" xml:space="preserve">ME, NS, CB parallelas eſſe; </s>
  <s xml:id="echoid-s13937" xml:space="preserve">_punctum_ M liberè ſumi; </s>
  <s xml:id="echoid-s13938" xml:space="preserve">_arcum_ <lb/>MN indefinitè parvum eſſe; </s>
  <s xml:id="echoid-s13939" xml:space="preserve">rectam α β curvæ VB, α μ curvæ AM, <lb/>μ ν _arcui_ MN æquales eſſe; </s>
  <s xml:id="echoid-s13940" xml:space="preserve">ad rectam α β applicatas ei perpendicu-<lb/>lares eſſe. </s>
  <s xml:id="echoid-s13941" xml:space="preserve">His præſtratis,</s>
</p>
<div xml:id="echoid-div434" type="float" level="2" n="1">
<note position="right" xlink:label="note-0283-01" xlink:href="note-0283-01a" xml:space="preserve">_Praparati@_ <lb/>_Communis_.</note>
</div>
<p>
  <s xml:id="echoid-s13942" xml:space="preserve">I. </s>
  <s xml:id="echoid-s13943" xml:space="preserve">Sit MP curvæ AB perpendicularis; </s>
  <s xml:id="echoid-s13944" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13945" xml:space="preserve">lineæ KZ L, α φ δta-<lb/>
<anchor type="note" xlink:label="note-0283-02a" xlink:href="note-0283-02"/>
les, ut FZ ipſi MP, &amp; </s>
  <s xml:id="echoid-s13946" xml:space="preserve">μ φ ipſi M Fæquentur; </s>
  <s xml:id="echoid-s13947" xml:space="preserve">erît _ſpatium_ α β δ ipſi <lb/>AD LK æquale.</s>
  <s xml:id="echoid-s13948" xml:space="preserve"/>
</p>
<div xml:id="echoid-div435" type="float" level="2" n="2">
<note position="right" xlink:label="note-0283-02" xlink:href="note-0283-02a" xml:space="preserve">Fig. 156, <lb/>157.</note>
</div>
<p>
  <s xml:id="echoid-s13949" xml:space="preserve">Nam _Triangula_ MRN, PFM ſimilia ſunt, adeoque MN. </s>
  <s xml:id="echoid-s13950" xml:space="preserve">NR <lb/>:</s>
  <s xml:id="echoid-s13951" xml:space="preserve">: PM. </s>
  <s xml:id="echoid-s13952" xml:space="preserve">MF. </s>
  <s xml:id="echoid-s13953" xml:space="preserve">unde MN x MF = NR x PM, hoc eſt (ſubſtitutis <lb/>æqualibus) μ ν x μ φ = FG x FZ; </s>
  <s xml:id="echoid-s13954" xml:space="preserve">ſeu rectang. </s>
  <s xml:id="echoid-s13955" xml:space="preserve">μ θ = rectang. </s>
  <s xml:id="echoid-s13956" xml:space="preserve">FH; <lb/></s>
  <s xml:id="echoid-s13957" xml:space="preserve">ſpatium verò α β δ minimè differt ab indeſinitè multis rectangulis, <lb/>qualia μθ &amp; </s>
  <s xml:id="echoid-s13958" xml:space="preserve">ſpatium AD LK totidem rectangulis, qualia FH, æ-<lb/>quivalet. </s>
  <s xml:id="echoid-s13959" xml:space="preserve">unde liquet Propoſitum.</s>
  <s xml:id="echoid-s13960" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13961" xml:space="preserve">II. </s>
  <s xml:id="echoid-s13962" xml:space="preserve">Hinc, ſi curva AMB circa axem AD rotetur, habebit ſe _pro._ <lb/></s>
  <s xml:id="echoid-s13963" xml:space="preserve">_ducta ſuperficies_ ad _ſpatium_ AD LK, ut _Circumferentia circuli Ad ra-_ <lb/>
<anchor type="note" xlink:label="note-0283-03a" xlink:href="note-0283-03"/>
_dium_; </s>
  <s xml:id="echoid-s13964" xml:space="preserve">unde noto ſpatio AD LK cognoſcetur dicta _ſuperficies._ </s>
  <s xml:id="echoid-s13965" xml:space="preserve">Con-<lb/>ſequentiæ rationem jam anteà pridem aſſignavimus.</s>
  <s xml:id="echoid-s13966" xml:space="preserve"/>
</p>
<div xml:id="echoid-div436" type="float" level="2" n="3">
<note position="right" xlink:label="note-0283-03" xlink:href="note-0283-03a" xml:space="preserve">Fig. 156.</note>
</div>
<p>
  <s xml:id="echoid-s13967" xml:space="preserve">III. </s>
  <s xml:id="echoid-s13968" xml:space="preserve">Exhinc _Spbæræ, Spbæroidis_ utriuſque, _Conidúmque ſuperficies_ <lb/>_dimenſionem_ accipiunt; </s>
  <s xml:id="echoid-s13969" xml:space="preserve">nam ſi AD ſit conicæ ſectionis, à qua iſtæ <lb/>figuræ oriuntur, axis; </s>
  <s xml:id="echoid-s13970" xml:space="preserve">linea KZL ſemper aliqua conicarum exiſtet, <lb/>haud difficili negotio determinabilis. </s>
  <s xml:id="echoid-s13971" xml:space="preserve">Hoc ſuggero tantùm, quoniam <lb/>nunc evulgatum habet ur.</s>
  <s xml:id="echoid-s13972" xml:space="preserve"/>
</p>
<pb o="106" file="0284" n="299" rhead=""/>
<p>
  <s xml:id="echoid-s13973" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s13974" xml:space="preserve">Iiſdem ſtantibus, ſit curva AYI talis, ut ordinata FY ſit in-<lb/>ter congruas FM, FZ proportione media; </s>
  <s xml:id="echoid-s13975" xml:space="preserve">erit _ſolidum_ ex ſpatio αδβ <lb/>
<anchor type="note" xlink:label="note-0284-01a" xlink:href="note-0284-01"/>
circa axem α β rotato factum æquale _ſolido_, quod à _ſpatio_ ADI circa <lb/>axem AD converſo procreatur.</s>
  <s xml:id="echoid-s13976" xml:space="preserve"/>
</p>
<div xml:id="echoid-div437" type="float" level="2" n="4">
<note position="left" xlink:label="note-0284-01" xlink:href="note-0284-01a" xml:space="preserve">Fig. 156, <lb/>157.</note>
</div>
<p>
  <s xml:id="echoid-s13977" xml:space="preserve">Nam eſt MN. </s>
  <s xml:id="echoid-s13978" xml:space="preserve">NR:</s>
  <s xml:id="echoid-s13979" xml:space="preserve">: PM. </s>
  <s xml:id="echoid-s13980" xml:space="preserve">MF:</s>
  <s xml:id="echoid-s13981" xml:space="preserve">: PM x MF. </s>
  <s xml:id="echoid-s13982" xml:space="preserve">MF q:</s>
  <s xml:id="echoid-s13983" xml:space="preserve">:FZ x <lb/>FM. </s>
  <s xml:id="echoid-s13984" xml:space="preserve">MFq. </s>
  <s xml:id="echoid-s13985" xml:space="preserve">unde MN x MFq = NR x FZ x FM; </s>
  <s xml:id="echoid-s13986" xml:space="preserve">hoc eſt <lb/>μ ν x μ φ q = NR x FYq. </s>
  <s xml:id="echoid-s13987" xml:space="preserve">Unde liquet Propoſitum.</s>
  <s xml:id="echoid-s13988" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s13989" xml:space="preserve">V. </s>
  <s xml:id="echoid-s13990" xml:space="preserve">Simili ratione colligetur, ſi FY ponatur inter FM, FZ _bime-_ <lb/>
<anchor type="note" xlink:label="note-0284-02a" xlink:href="note-0284-02"/>
_media_, fore _ſummam cuborum_ ex applicatis (quales μ φ) à curva α φ δ <lb/>ad rectam α β, æqualem _ſummæ cuborum_ ex explicatis à curva AYI ad <lb/>rectam AD. </s>
  <s xml:id="echoid-s13991" xml:space="preserve">paríque modo ſe res habebit quoad cæteras _poteſta-_ <lb/>_tes._</s>
  <s xml:id="echoid-s13992" xml:space="preserve"/>
</p>
<div xml:id="echoid-div438" type="float" level="2" n="5">
<note position="left" xlink:label="note-0284-02" xlink:href="note-0284-02a" xml:space="preserve">Fig. 156, <lb/>157.</note>
</div>
<p>
  <s xml:id="echoid-s13993" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s13994" xml:space="preserve">Porrò, ſtantibus reliquis, ſit curva VXO talis, ut EX ipſi MP <lb/>æquetur; </s>
  <s xml:id="echoid-s13995" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s13996" xml:space="preserve">curva πξψ talis, ut μ ξ æ quetur ipſi PF; </s>
  <s xml:id="echoid-s13997" xml:space="preserve">erit ſpatium <lb/>
<anchor type="note" xlink:label="note-0284-03a" xlink:href="note-0284-03"/>
α π ψ β æqua le ſpatio DV OB.</s>
  <s xml:id="echoid-s13998" xml:space="preserve"/>
</p>
<div xml:id="echoid-div439" type="float" level="2" n="6">
<note position="left" xlink:label="note-0284-03" xlink:href="note-0284-03a" xml:space="preserve">Fig. 156.</note>
</div>
<p>
  <s xml:id="echoid-s13999" xml:space="preserve">Nam eſt MN. </s>
  <s xml:id="echoid-s14000" xml:space="preserve">MR:</s>
  <s xml:id="echoid-s14001" xml:space="preserve">: MP. </s>
  <s xml:id="echoid-s14002" xml:space="preserve">PF; </s>
  <s xml:id="echoid-s14003" xml:space="preserve">adeoque MN x PF = MR <lb/>x MP. </s>
  <s xml:id="echoid-s14004" xml:space="preserve">hoc eſt μ ν x μ ξ = ES x EX. </s>
  <s xml:id="echoid-s14005" xml:space="preserve">vel rectang. </s>
  <s xml:id="echoid-s14006" xml:space="preserve">ET = rectang. <lb/></s>
  <s xml:id="echoid-s14007" xml:space="preserve">μ σ. </s>
  <s xml:id="echoid-s14008" xml:space="preserve">Unde liquet Propoſitum.</s>
  <s xml:id="echoid-s14009" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14010" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s14011" xml:space="preserve">Subnotetur hoc: </s>
  <s xml:id="echoid-s14012" xml:space="preserve">Si curva AB ſit _Parabola_, cujus _Axis_ AD, <lb/>
<anchor type="note" xlink:label="note-0284-04a" xlink:href="note-0284-04"/>
_parameter_ R; </s>
  <s xml:id="echoid-s14013" xml:space="preserve">erit curva VXO _byperbola_, cujus _centrum_ D, _Axis_ DV, <lb/>cujuſque _parameter_ axi R æquatur (ſcilicet ob EXq = (PMq = <lb/>PFq + FMq = {R q/4}+FMq = {R q/4}+ DEq = ) DVq+ DEq). <lb/></s>
  <s xml:id="echoid-s14014" xml:space="preserve">item _ſpatium_ α β ψ π erit _Rectangulum_; </s>
  <s xml:id="echoid-s14015" xml:space="preserve">quoniam ſingulæ applicatæ <lb/>μ ξ ipſi {R/2} æquantur. </s>
  <s xml:id="echoid-s14016" xml:space="preserve">Conſtat itaque dato _ſpatio byperbolico_ DVOB <lb/>curvam AMB dari; </s>
  <s xml:id="echoid-s14017" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14018" xml:space="preserve">viciſſim. </s>
  <s xml:id="echoid-s14019" xml:space="preserve">Hoc obiter.</s>
  <s xml:id="echoid-s14020" xml:space="preserve"/>
</p>
<div xml:id="echoid-div440" type="float" level="2" n="7">
<note position="left" xlink:label="note-0284-04" xlink:href="note-0284-04a" xml:space="preserve">Fig. 156.</note>
</div>
<p>
  <s xml:id="echoid-s14021" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s14022" xml:space="preserve">Adnotari poſſet etiam omnia ſimul quadrata ex applicatis <lb/>ad rectam α β à curva π ξ ψ æquari rectangulis omnibus ex PE, EX <lb/>
<anchor type="note" xlink:label="note-0284-05a" xlink:href="note-0284-05"/>
ad rectam DB applicatis (ſeu computatis); </s>
  <s xml:id="echoid-s14023" xml:space="preserve">cubos ex μ ξ æquari ipſis <lb/>PFq x EX; </s>
  <s xml:id="echoid-s14024" xml:space="preserve">ac ità porrò.</s>
  <s xml:id="echoid-s14025" xml:space="preserve"/>
</p>
<div xml:id="echoid-div441" type="float" level="2" n="8">
<note position="left" xlink:label="note-0284-05" xlink:href="note-0284-05a" xml:space="preserve">Fig. 157.</note>
</div>
<p>
  <s xml:id="echoid-s14026" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s14027" xml:space="preserve">Adjungatur etiam (productâ PM Q) ſi ponatur FZ æqua-<lb/>
<anchor type="note" xlink:label="note-0284-06a" xlink:href="note-0284-06"/>
lis ipſi PQ, &amp; </s>
  <s xml:id="echoid-s14028" xml:space="preserve">μ φ ipſi AQ; </s>
  <s xml:id="echoid-s14029" xml:space="preserve">_ſpatium_ α β δ _ſpatio_ AD LK æ-<lb/>quari.</s>
  <s xml:id="echoid-s14030" xml:space="preserve"/>
</p>
<div xml:id="echoid-div442" type="float" level="2" n="9">
<note position="left" xlink:label="note-0284-06" xlink:href="note-0284-06a" xml:space="preserve">Fig. 157.</note>
</div>
<pb o="107" file="0285" n="300" rhead=""/>
<p>
  <s xml:id="echoid-s14031" xml:space="preserve">Nam ob MN. </s>
  <s xml:id="echoid-s14032" xml:space="preserve">NR:</s>
  <s xml:id="echoid-s14033" xml:space="preserve">: PM. </s>
  <s xml:id="echoid-s14034" xml:space="preserve">MF:</s>
  <s xml:id="echoid-s14035" xml:space="preserve">: PQ. </s>
  <s xml:id="echoid-s14036" xml:space="preserve">QA; </s>
  <s xml:id="echoid-s14037" xml:space="preserve">erit MN x <lb/>QA = NR x QA; </s>
  <s xml:id="echoid-s14038" xml:space="preserve">hoc eſt rectang. </s>
  <s xml:id="echoid-s14039" xml:space="preserve">μ θ = rectang. </s>
  <s xml:id="echoid-s14040" xml:space="preserve">FH.</s>
  <s xml:id="echoid-s14041" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14042" xml:space="preserve">X. </s>
  <s xml:id="echoid-s14043" xml:space="preserve">Porrò, curvam AB tangat recta MT, ſintque curvæ DXO, <lb/>α φ δ tales, ut EX æquetur ipſi MT, &amp; </s>
  <s xml:id="echoid-s14044" xml:space="preserve">μ φ ipſi MF; </s>
  <s xml:id="echoid-s14045" xml:space="preserve">erit ſpatium <lb/>α β δ æquale _ſpatio_ DXOB.</s>
  <s xml:id="echoid-s14046" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve">Fig. 158. <lb/>159.</note>
<p>
  <s xml:id="echoid-s14047" xml:space="preserve">Nam MN. </s>
  <s xml:id="echoid-s14048" xml:space="preserve">MR:</s>
  <s xml:id="echoid-s14049" xml:space="preserve">: MT. </s>
  <s xml:id="echoid-s14050" xml:space="preserve">MF. </s>
  <s xml:id="echoid-s14051" xml:space="preserve">quare MN x MF = MR x MT; <lb/></s>
  <s xml:id="echoid-s14052" xml:space="preserve">hoc eſt μ ν x μφ = ES x EX; </s>
  <s xml:id="echoid-s14053" xml:space="preserve">unde patet.</s>
  <s xml:id="echoid-s14054" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14055" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s14056" xml:space="preserve">Hinc rurſus, _ſuperficies ſolidi ex ſpatii_ ABD circa axem AD <lb/>converſione progeniti ad _ſpatium_ DX OB ſe habet, ut _Circuli Cir-_ <lb/>
<anchor type="note" xlink:label="note-0285-02a" xlink:href="note-0285-02"/>
_cumf._ </s>
  <s xml:id="echoid-s14057" xml:space="preserve">ad _radium_; </s>
  <s xml:id="echoid-s14058" xml:space="preserve">hoc igitur noto ſimul illa innoteſcet. </s>
  <s xml:id="echoid-s14059" xml:space="preserve">unde rurſus <lb/>_Spbaroidum, Conoidumque ſuperficies_ dimetiri licebit.</s>
  <s xml:id="echoid-s14060" xml:space="preserve"/>
</p>
<div xml:id="echoid-div443" type="float" level="2" n="10">
<note position="right" xlink:label="note-0285-02" xlink:href="note-0285-02a" xml:space="preserve">Fig. 158.</note>
</div>
<p>
  <s xml:id="echoid-s14061" xml:space="preserve">XII. </s>
  <s xml:id="echoid-s14062" xml:space="preserve">Si linea DYI talis fuerit, ut ſit EY = √ EX x MF; </s>
  <s xml:id="echoid-s14063" xml:space="preserve">erit <lb/>_ſolidum_ ex _ſpatio_ αβδ circa axem αβ rotato factum æ quale _ſolido, quod_ <lb/>_ex ſpatio_ DBI circa axem DB rotato progignitur.</s>
  <s xml:id="echoid-s14064" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14065" xml:space="preserve">Etenim eſt MN. </s>
  <s xml:id="echoid-s14066" xml:space="preserve">MR:</s>
  <s xml:id="echoid-s14067" xml:space="preserve">: MT x MF. </s>
  <s xml:id="echoid-s14068" xml:space="preserve">MF q:</s>
  <s xml:id="echoid-s14069" xml:space="preserve">: EX x MF. </s>
  <s xml:id="echoid-s14070" xml:space="preserve">MFq <lb/>
<anchor type="note" xlink:label="note-0285-03a" xlink:href="note-0285-03"/>
:</s>
  <s xml:id="echoid-s14071" xml:space="preserve">: EYq. </s>
  <s xml:id="echoid-s14072" xml:space="preserve">MFq. </s>
  <s xml:id="echoid-s14073" xml:space="preserve">quare MN x MFq = MR x EYq. </s>
  <s xml:id="echoid-s14074" xml:space="preserve">hoc eſt μ ν <lb/>x μ φ q = ES x EYq.</s>
  <s xml:id="echoid-s14075" xml:space="preserve"/>
</p>
<div xml:id="echoid-div444" type="float" level="2" n="11">
<note position="right" xlink:label="note-0285-03" xlink:href="note-0285-03a" xml:space="preserve">Fig. 158. <lb/>159.</note>
</div>
<p>
  <s xml:id="echoid-s14076" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s14077" xml:space="preserve">Simili ratione _Cuborum (aliarumque poteſtatum)_ ex ordina-<lb/>tis μ φ _ſummas_ cum _ſpatiis_ ad rectam DB computatis licebit conferre.</s>
  <s xml:id="echoid-s14078" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14079" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s14080" xml:space="preserve">Sint prætereà lineæ AZK, αξψ ætales, ut FZ ipſi MT, &amp; </s>
  <s xml:id="echoid-s14081" xml:space="preserve"><lb/>μξ ipſi TF æquentur; </s>
  <s xml:id="echoid-s14082" xml:space="preserve">_ſpatium_ αβψ æquabitur _ſpatio_ ADK.</s>
  <s xml:id="echoid-s14083" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14084" xml:space="preserve">Etenim MN. </s>
  <s xml:id="echoid-s14085" xml:space="preserve">NR:</s>
  <s xml:id="echoid-s14086" xml:space="preserve">: MT. </s>
  <s xml:id="echoid-s14087" xml:space="preserve">TF; </s>
  <s xml:id="echoid-s14088" xml:space="preserve">hoc eſt μ ν. </s>
  <s xml:id="echoid-s14089" xml:space="preserve">FG:</s>
  <s xml:id="echoid-s14090" xml:space="preserve">: FZ. </s>
  <s xml:id="echoid-s14091" xml:space="preserve">μ ξ. <lb/></s>
  <s xml:id="echoid-s14092" xml:space="preserve">
<anchor type="note" xlink:label="note-0285-04a" xlink:href="note-0285-04"/>
quare μ ν x μ ξ = FG x FZ.</s>
  <s xml:id="echoid-s14093" xml:space="preserve"/>
</p>
<div xml:id="echoid-div445" type="float" level="2" n="12">
<note position="right" xlink:label="note-0285-04" xlink:href="note-0285-04a" xml:space="preserve">Fig. 158. <lb/>159.</note>
</div>
<p>
  <s xml:id="echoid-s14094" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s14095" xml:space="preserve">Etiam _ſumma quadratorum_ ex qpplicatis μ ξ æquatur _ſummæ_ <lb/>_Rectangulorum_ ex TF, FZ; </s>
  <s xml:id="echoid-s14096" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14097" xml:space="preserve">_ſumma Cuborum_ ex μ ξ æquantur <lb/>ipſis TFq x FZ (ad rectam ſcilicet AD computationem exigendo) <lb/>
<anchor type="note" xlink:label="note-0285-05a" xlink:href="note-0285-05"/>
paríque quoad cæteras poteſtates modò.</s>
  <s xml:id="echoid-s14098" xml:space="preserve"/>
</p>
<div xml:id="echoid-div446" type="float" level="2" n="13">
<note position="right" xlink:label="note-0285-05" xlink:href="note-0285-05a" xml:space="preserve">Fig. 158, <lb/>159.</note>
</div>
<p>
  <s xml:id="echoid-s14099" xml:space="preserve">XVI. </s>
  <s xml:id="echoid-s14100" xml:space="preserve">Rurſus ponatur recta QMP curvæ AMB perpendicularis; <lb/></s>
  <s xml:id="echoid-s14101" xml:space="preserve">ſitque recta β δ æqualis ipſi BD, &amp; </s>
  <s xml:id="echoid-s14102" xml:space="preserve">compleatur _Rectangulum_ αβδζ; </s>
  <s xml:id="echoid-s14103" xml:space="preserve"><lb/>tum curva KZL talis ſit, ut FZ ipſi QP æquetur; </s>
  <s xml:id="echoid-s14104" xml:space="preserve">erit _rectang._ </s>
  <s xml:id="echoid-s14105" xml:space="preserve">αβδζ <lb/>
<anchor type="note" xlink:label="note-0285-06a" xlink:href="note-0285-06"/>
æquale _ſpatio_ AD LK.</s>
  <s xml:id="echoid-s14106" xml:space="preserve"/>
</p>
<div xml:id="echoid-div447" type="float" level="2" n="14">
<note position="right" xlink:label="note-0285-06" xlink:href="note-0285-06a" xml:space="preserve">Fig. 160, <lb/>161.</note>
</div>
<p>
  <s xml:id="echoid-s14107" xml:space="preserve">Nam eſt MN. </s>
  <s xml:id="echoid-s14108" xml:space="preserve">NR:</s>
  <s xml:id="echoid-s14109" xml:space="preserve">: (PM. </s>
  <s xml:id="echoid-s14110" xml:space="preserve">MF:</s>
  <s xml:id="echoid-s14111" xml:space="preserve">:) PQIF. </s>
  <s xml:id="echoid-s14112" xml:space="preserve">quare MN <lb/>x IF = NR x PQ; </s>
  <s xml:id="echoid-s14113" xml:space="preserve">hoc eſt μν x μξ = FG x FZ. </s>
  <s xml:id="echoid-s14114" xml:space="preserve">unde patet.</s>
  <s xml:id="echoid-s14115" xml:space="preserve"/>
</p>
<pb o="108" file="0286" n="301" rhead=""/>
<p>
  <s xml:id="echoid-s14116" xml:space="preserve">Hinc noto ſpatio AK LD cognoſcetur curvæ AMB quantitas.</s>
  <s xml:id="echoid-s14117" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14118" xml:space="preserve">XVII. </s>
  <s xml:id="echoid-s14119" xml:space="preserve">Item, poſito rectam TMY contingere curvam AM B, fa-<lb/>
<anchor type="note" xlink:label="note-0286-01a" xlink:href="note-0286-01"/>
ctâque β γ = BC, completóque _Rectangulo_ αβγψ, ſit curva OXX <lb/>talis, ut FX ipſi TY æquetur; </s>
  <s xml:id="echoid-s14120" xml:space="preserve">erit _ſpatium_ (infinitè protenſum) <lb/>AD OX X æquale _Rectangulo_ αβγψ.</s>
  <s xml:id="echoid-s14121" xml:space="preserve"/>
</p>
<div xml:id="echoid-div448" type="float" level="2" n="15">
<note position="left" xlink:label="note-0286-01" xlink:href="note-0286-01a" xml:space="preserve">Fig. 160, <lb/>161.</note>
</div>
<p>
  <s xml:id="echoid-s14122" xml:space="preserve">Nam MN. </s>
  <s xml:id="echoid-s14123" xml:space="preserve">NR:</s>
  <s xml:id="echoid-s14124" xml:space="preserve">: YT. </s>
  <s xml:id="echoid-s14125" xml:space="preserve">DA; </s>
  <s xml:id="echoid-s14126" xml:space="preserve">hoc eſt μ ν. </s>
  <s xml:id="echoid-s14127" xml:space="preserve">FG:</s>
  <s xml:id="echoid-s14128" xml:space="preserve">: FX. </s>
  <s xml:id="echoid-s14129" xml:space="preserve">μ θ. </s>
  <s xml:id="echoid-s14130" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14131" xml:space="preserve"><lb/>μ ν x μ θ = FG x FX. </s>
  <s xml:id="echoid-s14132" xml:space="preserve">quare liquet.</s>
  <s xml:id="echoid-s14133" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14134" xml:space="preserve">Hinc rurſus, explorato _ſpatio_ ADOXX curva AMB innoteſcet,</s>
</p>
<p>
  <s xml:id="echoid-s14135" xml:space="preserve">XVIII. </s>
  <s xml:id="echoid-s14136" xml:space="preserve">Quin adſumptâ quâpiam determinatâ R, &amp; </s>
  <s xml:id="echoid-s14137" xml:space="preserve">factâ rectâ β δ <lb/>
<anchor type="note" xlink:label="note-0286-02a" xlink:href="note-0286-02"/>
= R; </s>
  <s xml:id="echoid-s14138" xml:space="preserve">ſi curva OX X talis lit, ut MF. </s>
  <s xml:id="echoid-s14139" xml:space="preserve">MP:</s>
  <s xml:id="echoid-s14140" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s14141" xml:space="preserve">FX; </s>
  <s xml:id="echoid-s14142" xml:space="preserve">erit _rectan-_ <lb/>_gulum_ αβδ ζ æquale _ſpatio_ ADOXX. </s>
  <s xml:id="echoid-s14143" xml:space="preserve">ac inde comperto hoc ſpatio, <lb/>curva prorſus innoteſcet.</s>
  <s xml:id="echoid-s14144" xml:space="preserve"/>
</p>
<div xml:id="echoid-div449" type="float" level="2" n="16">
<note position="left" xlink:label="note-0286-02" xlink:href="note-0286-02a" xml:space="preserve">Fig. 160, <lb/>161.</note>
</div>
<p>
  <s xml:id="echoid-s14145" xml:space="preserve">Nam MN. </s>
  <s xml:id="echoid-s14146" xml:space="preserve">NR:</s>
  <s xml:id="echoid-s14147" xml:space="preserve">: MP. </s>
  <s xml:id="echoid-s14148" xml:space="preserve">MF:</s>
  <s xml:id="echoid-s14149" xml:space="preserve">: FX. </s>
  <s xml:id="echoid-s14150" xml:space="preserve">R. </s>
  <s xml:id="echoid-s14151" xml:space="preserve">adeóque MR x R = <lb/>NR x FX; </s>
  <s xml:id="echoid-s14152" xml:space="preserve">ceu μν x μξ = FG x FX.</s>
  <s xml:id="echoid-s14153" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14154" xml:space="preserve">Complura talia poſſent adponi; </s>
  <s xml:id="echoid-s14155" xml:space="preserve">ſed vereor ut hæc nimis quam ſuffi-<lb/>cere videantur.</s>
  <s xml:id="echoid-s14156" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14157" xml:space="preserve">XIX. </s>
  <s xml:id="echoid-s14158" xml:space="preserve">Adnotetur ſaltem, hæc omnia æquè vera fore, nec abſimili-<lb/>ter oſtendi, poſito curvæ AMB convexa rectam AD ſpectare.</s>
  <s xml:id="echoid-s14159" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14160" xml:space="preserve">XX. </s>
  <s xml:id="echoid-s14161" xml:space="preserve">Ex oſtenſis autem _methodus_ facilis emergit _curvàs_ (θεωδημαγι-<lb/>κπς) _deſignandi_, quæ _dimenſionem_ admittunt qualem qualem; </s>
  <s xml:id="echoid-s14162" xml:space="preserve">nimirum <lb/>ità procedas.</s>
  <s xml:id="echoid-s14163" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14164" xml:space="preserve">Quamlibet (tibi quadantenùs notam) _aream trapeziam rectangu-_ <lb/>_lam_, duabus parallelis rectis AK, DL; </s>
  <s xml:id="echoid-s14165" xml:space="preserve">rectâ AD; </s>
  <s xml:id="echoid-s14166" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14167" xml:space="preserve">lineâ quâ-<lb/>
<anchor type="note" xlink:label="note-0286-03a" xlink:href="note-0286-03"/>
cunque KL _comprebenſam_ accipe sîs. </s>
  <s xml:id="echoid-s14168" xml:space="preserve">ad iſtam verò ſic referatur al-<lb/>tera ADEC, ut ductâ quâ cunque rectâ FH ad DL parallelâ (quæ <lb/>ſecet lineas AD, CE, KL punctis F, G, H) adſumptàque rectâ de-<lb/>terminatâ Z; </s>
  <s xml:id="echoid-s14169" xml:space="preserve">ſit _quadr atum_ ex FH æquale _quadratis_ ex FG, &amp; </s>
  <s xml:id="echoid-s14170" xml:space="preserve">Z. <lb/></s>
  <s xml:id="echoid-s14171" xml:space="preserve">
<anchor type="note" xlink:label="note-0286-04a" xlink:href="note-0286-04"/>
quinetiam ſit curva AIB talis, ut ad ipſam productâ rectâ GF I, ſit <lb/>_rectangulum_ ex Z, &amp; </s>
  <s xml:id="echoid-s14172" xml:space="preserve">FI æquale _ſpatio_ AFGC; </s>
  <s xml:id="echoid-s14173" xml:space="preserve">erit _rectangulum_ <lb/>ex Z, &amp; </s>
  <s xml:id="echoid-s14174" xml:space="preserve">_curva_ AB æquale _ſpatio_ AD LK.</s>
  <s xml:id="echoid-s14175" xml:space="preserve"/>
</p>
<div xml:id="echoid-div450" type="float" level="2" n="17">
<note position="left" xlink:label="note-0286-03" xlink:href="note-0286-03a" xml:space="preserve">Fig. 162.</note>
<note position="left" xlink:label="note-0286-04" xlink:href="note-0286-04a" xml:space="preserve">Fig. 163.</note>
</div>
<p>
  <s xml:id="echoid-s14176" xml:space="preserve">Æ què procedit methodus, etiamſi recta AK ponatur inſinita.</s>
  <s xml:id="echoid-s14177" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14178" xml:space="preserve">_Exemp_. </s>
  <s xml:id="echoid-s14179" xml:space="preserve">1. </s>
  <s xml:id="echoid-s14180" xml:space="preserve">Sit KL _rectalinea_; </s>
  <s xml:id="echoid-s14181" xml:space="preserve">erit curva CGE _Hyperbola._</s>
  <s xml:id="echoid-s14182" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Fig. 162.</note>
<p>
  <s xml:id="echoid-s14183" xml:space="preserve">2. </s>
  <s xml:id="echoid-s14184" xml:space="preserve">Sit linea KL _Arcus Circuli_, cujus _Centrum_ D; </s>
  <s xml:id="echoid-s14185" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14186" xml:space="preserve">AK <lb/>
<anchor type="note" xlink:label="note-0286-06a" xlink:href="note-0286-06"/>
<pb o="109" file="0287" n="302" rhead=""/>
= Z; </s>
  <s xml:id="echoid-s14187" xml:space="preserve">erit curva AGF _Circulus_; </s>
  <s xml:id="echoid-s14188" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14189" xml:space="preserve">curva AB = <lb/>{AD 2}+{DL/2 AK}arc. </s>
  <s xml:id="echoid-s14190" xml:space="preserve">KL.</s>
  <s xml:id="echoid-s14191" xml:space="preserve"/>
</p>
<div xml:id="echoid-div451" type="float" level="2" n="18">
<note position="left" xlink:label="note-0286-06" xlink:href="note-0286-06a" xml:space="preserve">Fig. 163.</note>
</div>
<p>
  <s xml:id="echoid-s14192" xml:space="preserve">3. </s>
  <s xml:id="echoid-s14193" xml:space="preserve">Sit linea KL _Hyperbola æquilatera_, cujus _Centrum_ A, &amp; </s>
  <s xml:id="echoid-s14194" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0287-01a" xlink:href="note-0287-01"/>
_Axis_ AK = Z; </s>
  <s xml:id="echoid-s14195" xml:space="preserve">erit CGE recta linea; </s>
  <s xml:id="echoid-s14196" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14197" xml:space="preserve">curva AB <lb/>_Parabola._</s>
  <s xml:id="echoid-s14198" xml:space="preserve"/>
</p>
<div xml:id="echoid-div452" type="float" level="2" n="19">
<note position="right" xlink:label="note-0287-01" xlink:href="note-0287-01a" xml:space="preserve">Fig. 164.</note>
</div>
<p>
  <s xml:id="echoid-s14199" xml:space="preserve">4. </s>
  <s xml:id="echoid-s14200" xml:space="preserve">Sit Linea KL _Parabola_ (cujus axis AD) erit curva CG E <lb/>
<anchor type="note" xlink:label="note-0287-02a" xlink:href="note-0287-02"/>
quoque _Parabola_; </s>
  <s xml:id="echoid-s14201" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14202" xml:space="preserve">curva AB _Paraboliformium_ quæ-<lb/>dam.</s>
  <s xml:id="echoid-s14203" xml:space="preserve"/>
</p>
<div xml:id="echoid-div453" type="float" level="2" n="20">
<note position="right" xlink:label="note-0287-02" xlink:href="note-0287-02a" xml:space="preserve">Fig. 163.</note>
</div>
<p>
  <s xml:id="echoid-s14204" xml:space="preserve">5. </s>
  <s xml:id="echoid-s14205" xml:space="preserve">Sit curva KL _Paraboliformis_ quædam inverſa, vel infini-<lb/>
<anchor type="note" xlink:label="note-0287-03a" xlink:href="note-0287-03"/>
ta (talis ſcilicet ut ſit FH = √{Z3/AF}) erit curva AB _Cyclo-_ <lb/>_is_, ad _circulum_ pertinens, cujus _Diameter_ ipſi Z æqua-<lb/>tur.</s>
  <s xml:id="echoid-s14206" xml:space="preserve"/>
</p>
<div xml:id="echoid-div454" type="float" level="2" n="21">
<note position="right" xlink:label="note-0287-03" xlink:href="note-0287-03a" xml:space="preserve">Fig. 165.</note>
</div>
<p>
  <s xml:id="echoid-s14207" xml:space="preserve">Elegantiora forſan _Exempla_ ipſe circumſpectans excogitabis.</s>
  <s xml:id="echoid-s14208" xml:space="preserve"/>
</p>
<pb o="110" file="0288" n="303" rhead=""/>
</div>
<div xml:id="echoid-div456" type="section" level="1" n="44">
<head xml:id="echoid-head47" xml:space="preserve">APPENDICULA 1.</head>
<p>
  <s xml:id="echoid-s14209" xml:space="preserve">HIc demùm etſi præter inſtitutum ſit particularia nunc attingere; <lb/></s>
  <s xml:id="echoid-s14210" xml:space="preserve">qualibus ſanè, hæc generalia conſequentibus, admodum pro-<lb/>clive foret turgidum Volumen compingere (_amico tamen morem ge-_ <lb/>_rens operâ_ dignum cenſenti) ſubtexam ad _Circuli Tangentes Secantéſq_; </s>
  <s xml:id="echoid-s14211" xml:space="preserve"><lb/>ſpectantia nonnulla, pleraque de ſuprà poſitis emergentia.</s>
  <s xml:id="echoid-s14212" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div457" type="section" level="1" n="45">
<head xml:id="echoid-head48" style="it" xml:space="preserve">Præparatio Communis.</head>
<note position="left" xml:space="preserve">Fig. 166.</note>
<p>
  <s xml:id="echoid-s14213" xml:space="preserve">Eſto _cirtuli Quadrans_ ACB, quam tangant rectæ AH, BG; </s>
  <s xml:id="echoid-s14214" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14215" xml:space="preserve"><lb/>in productis HA, AC ſumantur AK, CE ſingulæ pares _radio_ CA; <lb/></s>
  <s xml:id="echoid-s14216" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14217" xml:space="preserve">_aſymptotis_ AC, CZ per K deſcripta ſit _Hyperbola_ KZZ; </s>
  <s xml:id="echoid-s14218" xml:space="preserve">_aſymp-_ <lb/>_totis_ BC, BG per E _byperbola_ LEO. </s>
  <s xml:id="echoid-s14219" xml:space="preserve">Sumatur etiam in arcu AB <lb/>
<anchor type="note" xlink:label="note-0288-02a" xlink:href="note-0288-02"/>
_punctum arbitrarium_ M, per quod ducantur recta CMS (tangenti <lb/>AH occurrens in S) recta MT circulum tangens; </s>
  <s xml:id="echoid-s14220" xml:space="preserve">recta MFZ ad <lb/>BC parallela, recta MPL ad AC parallela. </s>
  <s xml:id="echoid-s14221" xml:space="preserve">Sit denuò recta α β æ-<lb/>qualis _arcui_ AB, &amp; </s>
  <s xml:id="echoid-s14222" xml:space="preserve">α μ arcui AM; </s>
  <s xml:id="echoid-s14223" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14224" xml:space="preserve">rectæ α γ, ξ μ π ψ rectæ α β <lb/>perpendiculares; </s>
  <s xml:id="echoid-s14225" xml:space="preserve">quarum α γ = AC; </s>
  <s xml:id="echoid-s14226" xml:space="preserve">μξ = AS; </s>
  <s xml:id="echoid-s14227" xml:space="preserve">μψ = CS; </s>
  <s xml:id="echoid-s14228" xml:space="preserve">μπ <lb/>= MP.</s>
  <s xml:id="echoid-s14229" xml:space="preserve"/>
</p>
<div xml:id="echoid-div457" type="float" level="2" n="1">
<note position="left" xlink:label="note-0288-02" xlink:href="note-0288-02a" xml:space="preserve">Fig. 167.</note>
</div>
<p>
  <s xml:id="echoid-s14230" xml:space="preserve">I. </s>
  <s xml:id="echoid-s14231" xml:space="preserve">Recta CS æquatur rectæ FZ; </s>
  <s xml:id="echoid-s14232" xml:space="preserve">adeoque _ſumma ſecantium ad_ <lb/>_arcum_ AM pertinentium, &amp; </s>
  <s xml:id="echoid-s14233" xml:space="preserve">ad rectam AC applicatarum æquatur <lb/>_ſpatio byperbolico_ AF ZK.</s>
  <s xml:id="echoid-s14234" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14235" xml:space="preserve">Eſt enim CF. </s>
  <s xml:id="echoid-s14236" xml:space="preserve">CA:</s>
  <s xml:id="echoid-s14237" xml:space="preserve">: (CM. </s>
  <s xml:id="echoid-s14238" xml:space="preserve">CS:</s>
  <s xml:id="echoid-s14239" xml:space="preserve">:) CA. </s>
  <s xml:id="echoid-s14240" xml:space="preserve">CS. </s>
  <s xml:id="echoid-s14241" xml:space="preserve">adeòque CF <lb/>x CS = CAq. </s>
  <s xml:id="echoid-s14242" xml:space="preserve">item CF x FZ = CA x AK = CAq. </s>
  <s xml:id="echoid-s14243" xml:space="preserve">ergo CS = FZ.</s>
  <s xml:id="echoid-s14244" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14245" xml:space="preserve">II. </s>
  <s xml:id="echoid-s14246" xml:space="preserve">_Spatium_ αμξ (hoc eſt _Smmma tangentium in arcu_ AM ad re-<lb/>
<anchor type="note" xlink:label="note-0288-03a" xlink:href="note-0288-03"/>
ctam αμ applicatarum) æquatur _ſpatio byperbolico_ AFZK.</s>
  <s xml:id="echoid-s14247" xml:space="preserve"/>
</p>
<div xml:id="echoid-div458" type="float" level="2" n="2">
<note position="left" xlink:label="note-0288-03" xlink:href="note-0288-03a" xml:space="preserve">Fig. 167.</note>
</div>
<p>
  <s xml:id="echoid-s14248" xml:space="preserve">Patet ex hujuſce Lectionis 9.</s>
  <s xml:id="echoid-s14249" xml:space="preserve"/>
</p>
<pb o="111" file="0289" n="304" rhead=""/>
<p>
  <s xml:id="echoid-s14250" xml:space="preserve">III. </s>
  <s xml:id="echoid-s14251" xml:space="preserve">Curva AX X talis ſit, ut PX ſecanti CS (vel CT) æquetur; <lb/></s>
  <s xml:id="echoid-s14252" xml:space="preserve">_ſpatium_ AC PX hoc eſt _Summa ſecamium ad arcum_ AM pertinen-<lb/>tium, &amp; </s>
  <s xml:id="echoid-s14253" xml:space="preserve">ad CB applicatarum) æquatur _duplo ſectori_ ACM.</s>
  <s xml:id="echoid-s14254" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14255" xml:space="preserve">Nam <anchor type="note" xlink:href="" symbol="(a)"/> _ſpatium_ AF MX _segmenti_ AFM _duplum_ eſt; </s>
  <s xml:id="echoid-s14256" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14257" xml:space="preserve">_rect-_
<anchor type="note" xlink:label="note-0289-01a" xlink:href="note-0289-01"/>
<anchor type="note" xlink:label="note-0289-02a" xlink:href="note-0289-02"/>
_angulum_ FC PM _Trianguli_ FCM. </s>
  <s xml:id="echoid-s14258" xml:space="preserve">ergo _totum ſpatium_ ACPX <lb/>totius _ſectoris_ ACM duplum eſt.</s>
  <s xml:id="echoid-s14259" xml:space="preserve"/>
</p>
<div xml:id="echoid-div459" type="float" level="2" n="3">
<note position="right" xlink:label="note-0289-01" xlink:href="note-0289-01a" xml:space="preserve">Fig. 166.</note>
<note symbol="(a)" position="right" xlink:label="note-0289-02" xlink:href="note-0289-02a" xml:space="preserve">10. Lect. <lb/>XI.</note>
</div>
<p>
  <s xml:id="echoid-s14260" xml:space="preserve">Etiam hoc è 16. </s>
  <s xml:id="echoid-s14261" xml:space="preserve">hujus duodecimæ Lectionis apertè conſtat.</s>
  <s xml:id="echoid-s14262" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14263" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s14264" xml:space="preserve">Curva CVV talis ſit, ut PV _Tangenti_ AS æquetur; </s>
  <s xml:id="echoid-s14265" xml:space="preserve">erit <lb/>_ſpatium_ CVP (hoc eſt _ſumma tangentium ad arcum_ AM _pertinen-_ <lb/>
<anchor type="note" xlink:label="note-0289-03a" xlink:href="note-0289-03"/>
_tium_, &amp; </s>
  <s xml:id="echoid-s14266" xml:space="preserve">ad rectam CB applicatarum) æquale _ſemiſſi quadrati ex_ <lb/>_ſubtenſa_ AM.</s>
  <s xml:id="echoid-s14267" xml:space="preserve"/>
</p>
<div xml:id="echoid-div460" type="float" level="2" n="4">
<note position="right" xlink:label="note-0289-03" xlink:href="note-0289-03a" xml:space="preserve">Fig. 166.</note>
</div>
<p>
  <s xml:id="echoid-s14268" xml:space="preserve">Manifeſtè conſectatur ex ſeptima undecimæ Lectionis.</s>
  <s xml:id="echoid-s14269" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14270" xml:space="preserve">V. </s>
  <s xml:id="echoid-s14271" xml:space="preserve">Acceptâ CQ = CP; </s>
  <s xml:id="echoid-s14272" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14273" xml:space="preserve">ductâ QO ad CE parallelâ (quæ <lb/>_byperbolæ_ LE occurrat in O) erit _ſpatium byperbolicum_ PL OQ du-<lb/>ctum in _radium_ CB (ſeu _cylindricum ad_ bafin PLOQ, altitudine <lb/>BC (duplum _ſummæ quadratorum_ ex rectis CS, ſeu PX ad _arcum_ <lb/>
<anchor type="note" xlink:label="note-0289-04a" xlink:href="note-0289-04"/>
AM pertinentibus, &amp; </s>
  <s xml:id="echoid-s14274" xml:space="preserve">ad rectam CB applicatis.</s>
  <s xml:id="echoid-s14275" xml:space="preserve"/>
</p>
<div xml:id="echoid-div461" type="float" level="2" n="5">
<note position="right" xlink:label="note-0289-04" xlink:href="note-0289-04a" xml:space="preserve">Fig. 166.</note>
</div>
<p>
  <s xml:id="echoid-s14276" xml:space="preserve">Nam quia PL. </s>
  <s xml:id="echoid-s14277" xml:space="preserve">QO:</s>
  <s xml:id="echoid-s14278" xml:space="preserve">: (BQ. </s>
  <s xml:id="echoid-s14279" xml:space="preserve">BP. </s>
  <s xml:id="echoid-s14280" xml:space="preserve">hoc eſt:</s>
  <s xml:id="echoid-s14281" xml:space="preserve">:) BC + CP. <lb/></s>
  <s xml:id="echoid-s14282" xml:space="preserve">BC - CP; </s>
  <s xml:id="echoid-s14283" xml:space="preserve">erit componendo PL + QO. </s>
  <s xml:id="echoid-s14284" xml:space="preserve">QO:</s>
  <s xml:id="echoid-s14285" xml:space="preserve">: 2 BC. </s>
  <s xml:id="echoid-s14286" xml:space="preserve">BC <lb/>- CP. </s>
  <s xml:id="echoid-s14287" xml:space="preserve">item eſt QO. </s>
  <s xml:id="echoid-s14288" xml:space="preserve">BC:</s>
  <s xml:id="echoid-s14289" xml:space="preserve">: BC. </s>
  <s xml:id="echoid-s14290" xml:space="preserve">BC + CP; </s>
  <s xml:id="echoid-s14291" xml:space="preserve">ergò (pares ra-<lb/>tiones adjungendo) eſt PL + QO. </s>
  <s xml:id="echoid-s14292" xml:space="preserve">QO + QO. </s>
  <s xml:id="echoid-s14293" xml:space="preserve">BC = 2 BC. </s>
  <s xml:id="echoid-s14294" xml:space="preserve"><lb/>BC - CP + BC. </s>
  <s xml:id="echoid-s14295" xml:space="preserve">BC + CP; </s>
  <s xml:id="echoid-s14296" xml:space="preserve">hoc eſt PL + QO. </s>
  <s xml:id="echoid-s14297" xml:space="preserve">BC:</s>
  <s xml:id="echoid-s14298" xml:space="preserve">: <lb/>2 BCq. </s>
  <s xml:id="echoid-s14299" xml:space="preserve">BCq - CPQ (hoc eſt:</s>
  <s xml:id="echoid-s14300" xml:space="preserve">:) 2 BCq. </s>
  <s xml:id="echoid-s14301" xml:space="preserve">PMq. </s>
  <s xml:id="echoid-s14302" xml:space="preserve">verùm <lb/>eſt PXq. </s>
  <s xml:id="echoid-s14303" xml:space="preserve">BCq:</s>
  <s xml:id="echoid-s14304" xml:space="preserve">: BCq. </s>
  <s xml:id="echoid-s14305" xml:space="preserve">PMq. </s>
  <s xml:id="echoid-s14306" xml:space="preserve">vel(antecedentes duplando)2 PXq. </s>
  <s xml:id="echoid-s14307" xml:space="preserve"><lb/>BCq:</s>
  <s xml:id="echoid-s14308" xml:space="preserve">: 2BCq.</s>
  <s xml:id="echoid-s14309" xml:space="preserve">PMq. </s>
  <s xml:id="echoid-s14310" xml:space="preserve">ergò PL + QO. </s>
  <s xml:id="echoid-s14311" xml:space="preserve">BC:</s>
  <s xml:id="echoid-s14312" xml:space="preserve">: 2 PXq. </s>
  <s xml:id="echoid-s14313" xml:space="preserve">BCq. </s>
  <s xml:id="echoid-s14314" xml:space="preserve">vel PL x BC + <lb/>QOxBC.</s>
  <s xml:id="echoid-s14315" xml:space="preserve">BCq:</s>
  <s xml:id="echoid-s14316" xml:space="preserve">:2PXq. </s>
  <s xml:id="echoid-s14317" xml:space="preserve">BCq. </s>
  <s xml:id="echoid-s14318" xml:space="preserve">quare PL x BC + QO x BC = 2PXq. </s>
  <s xml:id="echoid-s14319" xml:space="preserve"><lb/>itaque BC in omnes PL + QO ducta adæquat omnia totidem PXq. </s>
  <s xml:id="echoid-s14320" xml:space="preserve"><lb/>unde conſtat Propoſitum.</s>
  <s xml:id="echoid-s14321" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14322" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s14323" xml:space="preserve">Hinc ſpatium αγψμ (hoc eſt _ſumma ſecantium in arcu_ AM <lb/>
<anchor type="note" xlink:label="note-0289-05a" xlink:href="note-0289-05"/>
ad αβ applicatarum) æquatur _ſubduple ſpatio byperbolico_ PLOQ.</s>
  <s xml:id="echoid-s14324" xml:space="preserve"/>
</p>
<div xml:id="echoid-div462" type="float" level="2" n="6">
<note position="right" xlink:label="note-0289-05" xlink:href="note-0289-05a" xml:space="preserve">Fig. 167.</note>
</div>
<p>
  <s xml:id="echoid-s14325" xml:space="preserve">Nam ſumatur arcus MN indefinitê parvus, &amp; </s>
  <s xml:id="echoid-s14326" xml:space="preserve">huic æqualis recta μ ν, <lb/>ducatúrque recta NR ad AC parallela. </s>
  <s xml:id="echoid-s14327" xml:space="preserve">Eſtque MN. </s>
  <s xml:id="echoid-s14328" xml:space="preserve">MR:</s>
  <s xml:id="echoid-s14329" xml:space="preserve">: (MC. <lb/></s>
  <s xml:id="echoid-s14330" xml:space="preserve">CF:</s>
  <s xml:id="echoid-s14331" xml:space="preserve">: CS. </s>
  <s xml:id="echoid-s14332" xml:space="preserve">CA:</s>
  <s xml:id="echoid-s14333" xml:space="preserve">: PX. </s>
  <s xml:id="echoid-s14334" xml:space="preserve">CA:</s>
  <s xml:id="echoid-s14335" xml:space="preserve">:) PXq. </s>
  <s xml:id="echoid-s14336" xml:space="preserve">PX x CA. </s>
  <s xml:id="echoid-s14337" xml:space="preserve">adeóque <lb/>MN x PX x CA = MR x PXq. </s>
  <s xml:id="echoid-s14338" xml:space="preserve">ſeu μν x μψ x CA = MR x <lb/>PXq. </s>
  <s xml:id="echoid-s14339" xml:space="preserve">atqui (ex præcedente) omnium MR x PXq ſumma ſpatii <lb/>PL OQ in CA ducti ſubdupla eſt. </s>
  <s xml:id="echoid-s14340" xml:space="preserve">Ergò omnia totidem μν x μ ψ <lb/>in CA ducta eidem ſubduplo æquantur. </s>
  <s xml:id="echoid-s14341" xml:space="preserve">quare ſpatium αγψμ (om-
<pb o="112" file="0290" n="305" rhead=""/>
@ibus μνκμψ par) æquatur ſubduplo ſpatii PLOQ.</s>
  <s xml:id="echoid-s14342" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14343" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s14344" xml:space="preserve">Omnia quadrata ex rectis μψ (ad rectam αμ applicais) æquant <lb/>
<anchor type="note" xlink:label="note-0290-01a" xlink:href="note-0290-01"/>
CA x CP x PX(hoc eſt _parallelipipedum Baſe Rectangulo_ ACPD, <lb/>_Altitudine_ CS).</s>
  <s xml:id="echoid-s14345" xml:space="preserve"/>
</p>
<div xml:id="echoid-div463" type="float" level="2" n="7">
<note position="left" xlink:label="note-0290-01" xlink:href="note-0290-01a" xml:space="preserve">Fig. 167.</note>
</div>
<p>
  <s xml:id="echoid-s14346" xml:space="preserve">Hujus _Effati demonſtrationem_ (quanquam π&amp;</s>
  <s xml:id="echoid-s14347" xml:space="preserve">χΗ&amp;</s>
  <s xml:id="echoid-s14348" xml:space="preserve">ν) tranſilio; </s>
  <s xml:id="echoid-s14349" xml:space="preserve">quo-<lb/>niam aliud _Scbema_ diſcursúmque præ reliquis pleríſque longiuſculum <lb/>expoſcit; </s>
  <s xml:id="echoid-s14350" xml:space="preserve">neque rem tanti video.</s>
  <s xml:id="echoid-s14351" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14352" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s14353" xml:space="preserve">Curva AYY talis ſit, ut FY æquetur ipſi AS; </s>
  <s xml:id="echoid-s14354" xml:space="preserve">ductâ tum rectâ YI <lb/>
<anchor type="note" xlink:label="note-0290-02a" xlink:href="note-0290-02"/>
ad AC parallela, erit etiam _ſpatium_ AC IY YA (hoc eſt _ſumma_ <lb/>_Tangentium_ ad _arcum_ AM pertinentium, &amp; </s>
  <s xml:id="echoid-s14355" xml:space="preserve">ad rectam AC applica-<lb/>tarum, unà cum _rectangulo_ FCIY) æquale _ſubduplo ſpatio byperbo-_ <lb/>_lico_ PL OQ.</s>
  <s xml:id="echoid-s14356" xml:space="preserve"/>
</p>
<div xml:id="echoid-div464" type="float" level="2" n="8">
<note position="left" xlink:label="note-0290-02" xlink:href="note-0290-02a" xml:space="preserve">Fig. 166.</note>
</div>
<p>
  <s xml:id="echoid-s14357" xml:space="preserve">Nam _ſpatium_ α γ π μ <anchor type="note" xlink:href="" symbol="(a)"/> æquatur _rectangule_ ACPD; </s>
  <s xml:id="echoid-s14358" xml:space="preserve">hoc eſt
<anchor type="note" xlink:label="note-0290-03a" xlink:href="note-0290-03"/>
_rectangulo_ FC IY (nam eſt CA. </s>
  <s xml:id="echoid-s14359" xml:space="preserve">AS:</s>
  <s xml:id="echoid-s14360" xml:space="preserve">: CF. </s>
  <s xml:id="echoid-s14361" xml:space="preserve">FM; </s>
  <s xml:id="echoid-s14362" xml:space="preserve">vel CAFY:</s>
  <s xml:id="echoid-s14363" xml:space="preserve">: <lb/>CF. </s>
  <s xml:id="echoid-s14364" xml:space="preserve">CP. </s>
  <s xml:id="echoid-s14365" xml:space="preserve">adeoq; </s>
  <s xml:id="echoid-s14366" xml:space="preserve">CA x CP = FY x CF). </s>
  <s xml:id="echoid-s14367" xml:space="preserve">item ſpatium γπψ (hoc eſt omnes <lb/>
<anchor type="note" xlink:label="note-0290-04a" xlink:href="note-0290-04"/>
rectæ TF ad αε applicatæ, quotquot ad arcum AM pertinent) <anchor type="note" xlink:href="" symbol="(b)"/> æ-
quatur _ſpatio_ AFY; </s>
  <s xml:id="echoid-s14368" xml:space="preserve">ergo _ſpatium_ ACIYA æquatur _ſpatio_ αγψμ; <lb/></s>
  <s xml:id="echoid-s14369" xml:space="preserve">hoc eſt (ut mox oſtenſum) _ſemiſſi ſpatii byperbolici_ PL OQ.</s>
  <s xml:id="echoid-s14370" xml:space="preserve"/>
</p>
<div xml:id="echoid-div465" type="float" level="2" n="9">
<note symbol="a" position="left" xlink:label="note-0290-03" xlink:href="note-0290-03a" xml:space="preserve">1. Lect. <lb/>XII.</note>
<note symbol="(b)" position="left" xlink:label="note-0290-04" xlink:href="note-0290-04a" xml:space="preserve">14. Lect. <lb/>XII.</note>
</div>
<p>
  <s xml:id="echoid-s14371" xml:space="preserve">Aliter illud, (eíque connexa) dimenſus ſum, _boc præmiſſo Lem-_ <lb/>_mate._</s>
  <s xml:id="echoid-s14372" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14373" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s14374" xml:space="preserve">Sit _Hyperbola aquilatera_ (axes nempe pares habens) ERK ad <lb/>cujus axes CE D, CI; </s>
  <s xml:id="echoid-s14375" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14376" xml:space="preserve">ad hos ordinatæ KI, KD; </s>
  <s xml:id="echoid-s14377" xml:space="preserve">ſit item curvâ <lb/>
<anchor type="note" xlink:label="note-0290-05a" xlink:href="note-0290-05"/>
EVY talis, ut in _byperbola_ liberè ſumpto puncto R, ductâque recta <lb/>RVS ad DC parallelâ, ſint SR, CE, SV continuè proportiona-<lb/>les; </s>
  <s xml:id="echoid-s14378" xml:space="preserve">connexâ rectâ CK, erit _Spatium_ CE YI _Sectoris byperbolici_ <lb/>KCE duplum.</s>
  <s xml:id="echoid-s14379" xml:space="preserve"/>
</p>
<div xml:id="echoid-div466" type="float" level="2" n="10">
<note position="left" xlink:label="note-0290-05" xlink:href="note-0290-05a" xml:space="preserve">Fig. 168.</note>
</div>
<p>
  <s xml:id="echoid-s14380" xml:space="preserve">Nam ducatur RT _byperbolam_ tangens, &amp; </s>
  <s xml:id="echoid-s14381" xml:space="preserve">R Had CI parallela. <lb/></s>
  <s xml:id="echoid-s14382" xml:space="preserve">Eſtque CH. </s>
  <s xml:id="echoid-s14383" xml:space="preserve">CE:</s>
  <s xml:id="echoid-s14384" xml:space="preserve">: CE. </s>
  <s xml:id="echoid-s14385" xml:space="preserve">CT. </s>
  <s xml:id="echoid-s14386" xml:space="preserve">quare CT = SV; </s>
  <s xml:id="echoid-s14387" xml:space="preserve">vel HT = RV. </s>
  <s xml:id="echoid-s14388" xml:space="preserve"><lb/>itaque _Spatium_ ED KY duplum eſt _ſegmenti_ EDK. </s>
  <s xml:id="echoid-s14389" xml:space="preserve">item _rectangu-_ <lb/>
<anchor type="note" xlink:label="note-0290-06a" xlink:href="note-0290-06"/>
_lum_ IKDC _trianguli_ CDK duplum eſt; </s>
  <s xml:id="echoid-s14390" xml:space="preserve">ergo _reliquum ſpatium_ <lb/>CE YI _reliqui ſectoris_ ECK duplum eſt.</s>
  <s xml:id="echoid-s14391" xml:space="preserve"/>
</p>
<div xml:id="echoid-div467" type="float" level="2" n="11">
<note position="left" xlink:label="note-0290-06" xlink:href="note-0290-06a" xml:space="preserve">10 Lect. XI.</note>
</div>
<p>
  <s xml:id="echoid-s14392" xml:space="preserve">X. </s>
  <s xml:id="echoid-s14393" xml:space="preserve">Reſumptâ jam quadrante circulari AC B, ſit CE = CA; <lb/></s>
  <s xml:id="echoid-s14394" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14395" xml:space="preserve">axe AE, _parametro etiam_ AE, deſcripta ſit _Hyperbola_ EKK; </s>
  <s xml:id="echoid-s14396" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0290-07a" xlink:href="note-0290-07"/>
poſitóque curvam AYY talem eſſe, ut ordinatâ quâcunque rectâ <lb/>MFY, ſit FY tangenti AS æqualis; </s>
  <s xml:id="echoid-s14397" xml:space="preserve">ducatur recta YIK (rectam
<pb o="113" file="0291" n="306" rhead=""/>
C 2 ſecans in I, _byperbolam_ in K) &amp; </s>
  <s xml:id="echoid-s14398" xml:space="preserve">connectatur CK; </s>
  <s xml:id="echoid-s14399" xml:space="preserve">erit ſpatium <lb/>ACIYA _ſectoris byperbolici_ ECK duplum.</s>
  <s xml:id="echoid-s14400" xml:space="preserve"/>
</p>
<div xml:id="echoid-div468" type="float" level="2" n="12">
<note position="left" xlink:label="note-0290-07" xlink:href="note-0290-07a" xml:space="preserve">Fig. 169.</note>
</div>
<p>
  <s xml:id="echoid-s14401" xml:space="preserve">Nam eſt CIq. </s>
  <s xml:id="echoid-s14402" xml:space="preserve">CAq :</s>
  <s xml:id="echoid-s14403" xml:space="preserve">: ASq. </s>
  <s xml:id="echoid-s14404" xml:space="preserve">CAq:</s>
  <s xml:id="echoid-s14405" xml:space="preserve">: FMq. </s>
  <s xml:id="echoid-s14406" xml:space="preserve">CFq:</s>
  <s xml:id="echoid-s14407" xml:space="preserve">: CAq <lb/>
<anchor type="note" xlink:label="note-0291-01a" xlink:href="note-0291-01"/>
- CFq. </s>
  <s xml:id="echoid-s14408" xml:space="preserve">CFq. </s>
  <s xml:id="echoid-s14409" xml:space="preserve">componendóque CIq + CAq. </s>
  <s xml:id="echoid-s14410" xml:space="preserve">CAq:</s>
  <s xml:id="echoid-s14411" xml:space="preserve">: <lb/>CAq. </s>
  <s xml:id="echoid-s14412" xml:space="preserve">CFq. </s>
  <s xml:id="echoid-s14413" xml:space="preserve">hoc eſt (ex _byperbolœ_ natura) IKq. </s>
  <s xml:id="echoid-s14414" xml:space="preserve">CAq:</s>
  <s xml:id="echoid-s14415" xml:space="preserve">: CAq. <lb/></s>
  <s xml:id="echoid-s14416" xml:space="preserve">CFq. </s>
  <s xml:id="echoid-s14417" xml:space="preserve">vel IK. </s>
  <s xml:id="echoid-s14418" xml:space="preserve">CE :</s>
  <s xml:id="echoid-s14419" xml:space="preserve">: CE. </s>
  <s xml:id="echoid-s14420" xml:space="preserve">IY. </s>
  <s xml:id="echoid-s14421" xml:space="preserve">itaque _ſpatium_ ACIYA _ſectoris_ <lb/>ECK duplum eſſe perſpicuum eſt è præcedente.</s>
  <s xml:id="echoid-s14422" xml:space="preserve"/>
</p>
<div xml:id="echoid-div469" type="float" level="2" n="13">
<note position="right" xlink:label="note-0291-01" xlink:href="note-0291-01a" xml:space="preserve">Fig. 169.</note>
</div>
<p>
  <s xml:id="echoid-s14423" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s14424" xml:space="preserve">_Coroll_. </s>
  <s xml:id="echoid-s14425" xml:space="preserve">Hinc ſi Polo E, _Cbordà_ CB, _Sagittâ_ CAdeſcripta ſit <lb/>_Concbois_ AVV, cui occurrat YFM producta in V; </s>
  <s xml:id="echoid-s14426" xml:space="preserve">erit MV = FY; <lb/></s>
  <s xml:id="echoid-s14427" xml:space="preserve">adeóque _ſpatium_ AMV _ſpatio_ AFY æquatur.</s>
  <s xml:id="echoid-s14428" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14429" xml:space="preserve">XII.</s>
  <s xml:id="echoid-s14430" xml:space="preserve">Unde _ſpatiorum_ ejuſmodi _Conchoidalium dim@nſiones_ innoteſcunt.</s>
  <s xml:id="echoid-s14431" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14432" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s14433" xml:space="preserve">Neſcio, an _operæ_ ſit hoc adjicere _Corollarium_.</s>
  <s xml:id="echoid-s14434" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14435" xml:space="preserve">XIII. </s>
  <s xml:id="echoid-s14436" xml:space="preserve">Sit recta AErectæ RSperpendicularis; </s>
  <s xml:id="echoid-s14437" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14438" xml:space="preserve">CE = CA; <lb/></s>
  <s xml:id="echoid-s14439" xml:space="preserve">ſintque duæ (ſibimet inverſæ) _conchoides_ AZZ, EYY ad eundem <lb/>_polum_ E, _communémque regulam_ RS deſcriptæ, ab E verò ducatur <lb/>
<anchor type="note" xlink:label="note-0291-02a" xlink:href="note-0291-02"/>
utcunque recta EYZ (lineas interſecans, ut vides) ſit etiam _byperbole_ <lb/>_œquilatera_, EKK, cujus _centrum_ C, _ſemiaxis_ CE; </s>
  <s xml:id="echoid-s14440" xml:space="preserve">du&amp;</s>
  <s xml:id="echoid-s14441" xml:space="preserve">âque IK <lb/>ad AE parallelâ, connectatur CK, erit _ſpatium quadrilineum_ <lb/>AEOYZPA (rectis AE, YZ, &amp; </s>
  <s xml:id="echoid-s14442" xml:space="preserve">_concbis_ EOY, APZ compre-<lb/>henſum) æquale _quadruplo ſectori Hyperbolico_ ECK.</s>
  <s xml:id="echoid-s14443" xml:space="preserve"/>
</p>
<div xml:id="echoid-div470" type="float" level="2" n="14">
<note position="right" xlink:label="note-0291-02" xlink:href="note-0291-02a" xml:space="preserve">Fig. 170.</note>
</div>
<p>
  <s xml:id="echoid-s14444" xml:space="preserve">Nam ſi _centro_ E per C ducatur _arcus circularis_ CX; </s>
  <s xml:id="echoid-s14445" xml:space="preserve">è dictis faci-<lb/>lè colligetur _ſpatium_ APZIC æquari _duplo ſectori hyperbolico_ ECK <lb/>unà cum _ſectore circulari_ CEX. </s>
  <s xml:id="echoid-s14446" xml:space="preserve">item _ſpatium_ EOYIC æquari _duplo_ <lb/>_ſectori_ ECK, _dempto ſectore_ CEX.</s>
  <s xml:id="echoid-s14447" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14448" xml:space="preserve">Ità quoque facile colligas. </s>
  <s xml:id="echoid-s14449" xml:space="preserve">Ducantur ZF, YGad CS parallelæ; <lb/></s>
  <s xml:id="echoid-s14450" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14451" xml:space="preserve">protrahantur GYL, LIH. </s>
  <s xml:id="echoid-s14452" xml:space="preserve">ac ob IY = IZ, eſt FZ + GY = <lb/>2 CI. </s>
  <s xml:id="echoid-s14453" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14454" xml:space="preserve">_trapezium_ FGYZ = _rectang._ </s>
  <s xml:id="echoid-s14455" xml:space="preserve">EGLH = 2 CG x CI. </s>
  <s xml:id="echoid-s14456" xml:space="preserve"><lb/>ergò patet.</s>
  <s xml:id="echoid-s14457" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14458" xml:space="preserve">Adnotari poteſt, ſi lubet, ductâ ATad CSparallelâ, protractâ-<lb/>que EZT, ſi ponatur N = 2 triang. </s>
  <s xml:id="echoid-s14459" xml:space="preserve">CEI - 2 ſect. </s>
  <s xml:id="echoid-s14460" xml:space="preserve">ECK; </s>
  <s xml:id="echoid-s14461" xml:space="preserve">fore <lb/>ſpat EZT + EOYE = 2 N.</s>
  <s xml:id="echoid-s14462" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14463" xml:space="preserve">Nempe N + CXI = ſpat. </s>
  <s xml:id="echoid-s14464" xml:space="preserve">AZT. </s>
  <s xml:id="echoid-s14465" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14466" xml:space="preserve">N - CXI = ſpat. <lb/></s>
  <s xml:id="echoid-s14467" xml:space="preserve">EOY E.</s>
  <s xml:id="echoid-s14468" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14469" xml:space="preserve">XIV. </s>
  <s xml:id="echoid-s14470" xml:space="preserve">Adjiciemus etiam hiſce cognatam _Ciſſoidalis ſpatii_ dimenſio-<lb/>nem.</s>
  <s xml:id="echoid-s14471" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14472" xml:space="preserve">Sit _Semicirculus_ AMB (cujus centrum C) quem tangat recta <lb/>
<anchor type="note" xlink:label="note-0291-03a" xlink:href="note-0291-03"/>
AH; </s>
  <s xml:id="echoid-s14473" xml:space="preserve">eique congruens _Ciſſois_ AZZ cujus ſcilicet hæc proprietas eſt,
<pb o="114" file="0292" n="307" rhead=""/>
_utin circumf._ </s>
  <s xml:id="echoid-s14474" xml:space="preserve">AMB ſumpto utcunque puncto M, &amp; </s>
  <s xml:id="echoid-s14475" xml:space="preserve">per hoc trajectâ <lb/>
<anchor type="note" xlink:label="note-0292-01a" xlink:href="note-0292-01"/>
rectâ BMZ, ductâque rectâ MFZ, quæ curvam AZZ ſecet in Z, <lb/>ſit MZ = AS) in recta verò α β ſumatur αμ æqualis arcui AM, &amp; </s>
  <s xml:id="echoid-s14476" xml:space="preserve"><lb/>ad αμ applicentur rectæ perpendiculares μ ξ æquales _arcunm_ AMſinu-<lb/>_bus verſis_ AF; </s>
  <s xml:id="echoid-s14477" xml:space="preserve">erit _ſpatium trilineum_ MAZ _ſpatii αμξ duplum._</s>
  <s xml:id="echoid-s14478" xml:space="preserve"/>
</p>
<div xml:id="echoid-div471" type="float" level="2" n="15">
<note position="right" xlink:label="note-0291-03" xlink:href="note-0291-03a" xml:space="preserve">Fig. 171.</note>
<note position="left" xlink:label="note-0292-01" xlink:href="note-0292-01a" xml:space="preserve">Fig. 171.</note>
</div>
<p>
  <s xml:id="echoid-s14479" xml:space="preserve">Nam ſumatur _arcus_ MNindeſinitè parvus, &amp; </s>
  <s xml:id="echoid-s14480" xml:space="preserve">ei æqualis μν; </s>
  <s xml:id="echoid-s14481" xml:space="preserve">du-<lb/>catúrque recta NRad ABparallela, connectatúrque recta CM. </s>
  <s xml:id="echoid-s14482" xml:space="preserve">Eſt-<lb/>que jam AS. </s>
  <s xml:id="echoid-s14483" xml:space="preserve">AB (2 CM):</s>
  <s xml:id="echoid-s14484" xml:space="preserve">: (FM. </s>
  <s xml:id="echoid-s14485" xml:space="preserve">FB:</s>
  <s xml:id="echoid-s14486" xml:space="preserve">:) AF. </s>
  <s xml:id="echoid-s14487" xml:space="preserve">FM. </s>
  <s xml:id="echoid-s14488" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14489" xml:space="preserve">2 CM. <lb/></s>
  <s xml:id="echoid-s14490" xml:space="preserve">2 MN:</s>
  <s xml:id="echoid-s14491" xml:space="preserve">: CM. </s>
  <s xml:id="echoid-s14492" xml:space="preserve">MN:</s>
  <s xml:id="echoid-s14493" xml:space="preserve">:) FM. </s>
  <s xml:id="echoid-s14494" xml:space="preserve">NR. </s>
  <s xml:id="echoid-s14495" xml:space="preserve">quapropter erit ex æquo AS. </s>
  <s xml:id="echoid-s14496" xml:space="preserve"><lb/>2 MN:</s>
  <s xml:id="echoid-s14497" xml:space="preserve">: AF. </s>
  <s xml:id="echoid-s14498" xml:space="preserve">NR; </s>
  <s xml:id="echoid-s14499" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14500" xml:space="preserve">ideò NR x AS = 2 MN x AF. </s>
  <s xml:id="echoid-s14501" xml:space="preserve">hoc eſt <lb/>NR x MZ = 2 μν x μξ. </s>
  <s xml:id="echoid-s14502" xml:space="preserve">unde _ſpatium_ MAZ _duplo ſpatio_ α μξ æ-<lb/>quatur.</s>
  <s xml:id="echoid-s14503" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14504" xml:space="preserve">Hinc cum _ſpatii_ αμξ dimenſio vulgò nota ſit, &amp; </s>
  <s xml:id="echoid-s14505" xml:space="preserve">è ſuprà poſitis <lb/>etiam facilè deducatur; </s>
  <s xml:id="echoid-s14506" xml:space="preserve">habetur _ſpatii ciſſoidalis_ MAZ _dimenſio._ </s>
  <s xml:id="echoid-s14507" xml:space="preserve">cal-<lb/>
<anchor type="note" xlink:label="note-0292-02a" xlink:href="note-0292-02"/>
culum ineat qui volet.</s>
  <s xml:id="echoid-s14508" xml:space="preserve"/>
</p>
<div xml:id="echoid-div472" type="float" level="2" n="16">
<note position="left" xlink:label="note-0292-02" xlink:href="note-0292-02a" xml:space="preserve">Fig 172.</note>
</div>
<p>
  <s xml:id="echoid-s14509" xml:space="preserve">Iſta claudet hoc _Conſectariolum:_</s>
  <s xml:id="echoid-s14510" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14511" xml:space="preserve">XV. </s>
  <s xml:id="echoid-s14512" xml:space="preserve">Sit _circuli quadrans_ ACB, _circulúmque_ tangant AH, BG; <lb/></s>
  <s xml:id="echoid-s14513" xml:space="preserve">ſintque curvæ KZZ, LEO _byperbolœ_, eædem quæ <anchor type="note" xlink:href="" symbol="(_a_)"/> ſuperiùs. </s>
  <s xml:id="echoid-s14514" xml:space="preserve">ar-
<anchor type="note" xlink:label="note-0292-03a" xlink:href="note-0292-03"/>
<anchor type="note" xlink:label="note-0292-04a" xlink:href="note-0292-04"/>
cus verò ſumptus AMin partes diviſus concipiatur indefinitè multas <lb/>punctis N; </s>
  <s xml:id="echoid-s14515" xml:space="preserve">per quæ trajiciantur radii CN; </s>
  <s xml:id="echoid-s14516" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14517" xml:space="preserve">his occurrant rectæ <lb/>NXad puncta X; </s>
  <s xml:id="echoid-s14518" xml:space="preserve">_ſumma rectarum_ NX(in radiis) æquatur ſpatio <lb/>{AFZK/Rad}; </s>
  <s xml:id="echoid-s14519" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14520" xml:space="preserve">_ſummarectarum_ NX (in parallelis ad AS) æquatur _ſpatio_ <lb/>{PLQO/3 Rad.</s>
  <s xml:id="echoid-s14521" xml:space="preserve">}.</s>
  <s xml:id="echoid-s14522" xml:space="preserve"/>
</p>
<div xml:id="echoid-div473" type="float" level="2" n="17">
<note position="left" xlink:label="note-0292-03" xlink:href="note-0292-03a" xml:space="preserve">Fig. 173.</note>
<note symbol="(_a_)" position="left" xlink:label="note-0292-04" xlink:href="note-0292-04a" xml:space="preserve">7, &amp; 12.</note>
</div>
<p>
  <s xml:id="echoid-s14523" xml:space="preserve">Nam triangulum XMN triangulo SAC ſimile eſt; </s>
  <s xml:id="echoid-s14524" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14525" xml:space="preserve">inde XM. <lb/></s>
  <s xml:id="echoid-s14526" xml:space="preserve">MN:</s>
  <s xml:id="echoid-s14527" xml:space="preserve">: AS. </s>
  <s xml:id="echoid-s14528" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s14529" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14530" xml:space="preserve">XN. </s>
  <s xml:id="echoid-s14531" xml:space="preserve">MN:</s>
  <s xml:id="echoid-s14532" xml:space="preserve">: CS. </s>
  <s xml:id="echoid-s14533" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s14534" xml:space="preserve">unde XM = <lb/>{MN x AS/CA}; </s>
  <s xml:id="echoid-s14535" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14536" xml:space="preserve">XN = {MN x CS/CA}. </s>
  <s xml:id="echoid-s14537" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14538" xml:space="preserve">ità in reliquis; </s>
  <s xml:id="echoid-s14539" xml:space="preserve">unde liquet <lb/>Proſitum, ex 2, &amp; </s>
  <s xml:id="echoid-s14540" xml:space="preserve">7 harum.</s>
  <s xml:id="echoid-s14541" xml:space="preserve"/>
</p>
<pb o="115" file="0293" n="308" rhead=""/>
</div>
<div xml:id="echoid-div475" type="section" level="1" n="46">
<head xml:id="echoid-head49" xml:space="preserve">APPENDICULA 2.</head>
<p>
  <s xml:id="echoid-s14542" xml:space="preserve">B Revitati ſimul ac perſpicuitati (huic autem præcipuè) conſulentes <lb/>præcedentia recto diſcurſu comprobata dedimus; </s>
  <s xml:id="echoid-s14543" xml:space="preserve">qualinon mo-<lb/>do veritas, opinor, ſatis ſirmatur, at ejuſdem origo limpidiùs appa-<lb/>ret. </s>
  <s xml:id="echoid-s14544" xml:space="preserve">Verùm nè quis, minùs hujuſmodi ratiociniis adſuetus, hæreat, <lb/>iſta paucula ſubdemus, quibus tales diſcurſus communiantur, quorum-<lb/>que ſubſidio non difficilè conficiantur _Propoſitorum demonſtrationes a-_ <lb/>_pagogicœ_.</s>
  <s xml:id="echoid-s14545" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14546" xml:space="preserve">I. </s>
  <s xml:id="echoid-s14547" xml:space="preserve">Sint quotlibet _rationes_ A ad X, B ad Y, C ad Z, ſingulæ deſig-<lb/>natâ quâ piam ratione R ad S majores; </s>
  <s xml:id="echoid-s14548" xml:space="preserve">erit _omnium antecedentium_ <lb/>(ſimul acceptarum) ad _omnes conſequentes ratio_ major ratione <lb/>R ad S.</s>
  <s xml:id="echoid-s14549" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve"> <lb/>A. X. # A. M. <lb/>B. Y. # B. N. <lb/>C. Z. # C. O. <lb/></note>
<p>
  <s xml:id="echoid-s14550" xml:space="preserve">Nam ſint rationes A ad M, B ad N, C ad O ſingulæ æquales ra-<lb/>tioni R ad S. </s>
  <s xml:id="echoid-s14551" xml:space="preserve">ergò X &amp;</s>
  <s xml:id="echoid-s14552" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s14553" xml:space="preserve">M; </s>
  <s xml:id="echoid-s14554" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14555" xml:space="preserve">Y &amp;</s>
  <s xml:id="echoid-s14556" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s14557" xml:space="preserve">N; </s>
  <s xml:id="echoid-s14558" xml:space="preserve">ac Z &amp;</s>
  <s xml:id="echoid-s14559" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s14560" xml:space="preserve">O. </s>
  <s xml:id="echoid-s14561" xml:space="preserve">patet igitur <lb/>fore A + B + C. </s>
  <s xml:id="echoid-s14562" xml:space="preserve">X + Y + Z &amp;</s>
  <s xml:id="echoid-s14563" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s14564" xml:space="preserve">A + B + C. </s>
  <s xml:id="echoid-s14565" xml:space="preserve">M + N + O. <lb/></s>
  <s xml:id="echoid-s14566" xml:space="preserve">hoc eſt A + B + C. </s>
  <s xml:id="echoid-s14567" xml:space="preserve">X + Y + Z &amp;</s>
  <s xml:id="echoid-s14568" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s14569" xml:space="preserve">R. </s>
  <s xml:id="echoid-s14570" xml:space="preserve">S.</s>
  <s xml:id="echoid-s14571" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14572" xml:space="preserve">II. </s>
  <s xml:id="echoid-s14573" xml:space="preserve">Hinc patet, ſi quotlibet rationes ſingulæ deſignabili quâcunque <lb/>majores ſint, _antecedentium ſummam ad ſummam conſequentium_ eti-<lb/>am deſignabili quâcunque majorem rationem habere.</s>
  <s xml:id="echoid-s14574" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14575" xml:space="preserve">III. </s>
  <s xml:id="echoid-s14576" xml:space="preserve">Sit curva quævis ADB, cujus axis AD, &amp; </s>
  <s xml:id="echoid-s14577" xml:space="preserve">ad hunc applica-
<pb o="116" file="0294" n="309" rhead=""/>
ta recta BD; </s>
  <s xml:id="echoid-s14578" xml:space="preserve">curvam verò tangat recta BT; </s>
  <s xml:id="echoid-s14579" xml:space="preserve">ſitque BP rectæ BD <lb/>particula indefinitè parva; </s>
  <s xml:id="echoid-s14580" xml:space="preserve">ducatúrque recta POad DTparallela, <lb/>
<anchor type="note" xlink:label="note-0294-01a" xlink:href="note-0294-01"/>
curvam ſecans ad N; </s>
  <s xml:id="echoid-s14581" xml:space="preserve">dico PNad NOrationem habere majorem quâ-<lb/>vis deſignabili, puta quàm R ad S.</s>
  <s xml:id="echoid-s14582" xml:space="preserve"/>
</p>
<div xml:id="echoid-div475" type="float" level="2" n="1">
<note position="left" xlink:label="note-0294-01" xlink:href="note-0294-01a" xml:space="preserve">Fig. 174.</note>
</div>
<p>
  <s xml:id="echoid-s14583" xml:space="preserve">Nam ſit DE. </s>
  <s xml:id="echoid-s14584" xml:space="preserve">ET:</s>
  <s xml:id="echoid-s14585" xml:space="preserve">: RS; </s>
  <s xml:id="echoid-s14586" xml:space="preserve">connexaque recta BEcurvam ſecet in <lb/>G, rectam POin K; </s>
  <s xml:id="echoid-s14587" xml:space="preserve">per G verò ducatur FHad DAparallela. <lb/></s>
  <s xml:id="echoid-s14588" xml:space="preserve">quoniam igitur BP ponitur indefinitè parva, eſt BP &amp;</s>
  <s xml:id="echoid-s14589" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s14590" xml:space="preserve">BF; </s>
  <s xml:id="echoid-s14591" xml:space="preserve">adeóq; </s>
  <s xml:id="echoid-s14592" xml:space="preserve"><lb/>PK &amp;</s>
  <s xml:id="echoid-s14593" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s14594" xml:space="preserve">PN (nam ſubtenſa BGintra curvam tota cadit). </s>
  <s xml:id="echoid-s14595" xml:space="preserve">ergo PN. </s>
  <s xml:id="echoid-s14596" xml:space="preserve"><lb/>NO &amp;</s>
  <s xml:id="echoid-s14597" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s14598" xml:space="preserve">PK. </s>
  <s xml:id="echoid-s14599" xml:space="preserve">KO:</s>
  <s xml:id="echoid-s14600" xml:space="preserve">: DE. </s>
  <s xml:id="echoid-s14601" xml:space="preserve">ET:</s>
  <s xml:id="echoid-s14602" xml:space="preserve">: R.</s>
  <s xml:id="echoid-s14603" xml:space="preserve">S.</s>
  <s xml:id="echoid-s14604" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14605" xml:space="preserve">IV. </s>
  <s xml:id="echoid-s14606" xml:space="preserve">Hinc, ſi baſis DBin partes ſecetur indeſinitè multas ad puncta <lb/>Z; </s>
  <s xml:id="echoid-s14607" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14608" xml:space="preserve">per hæc ducantur rectæ ad DAparallelæ curvam ſecantes pun-<lb/>ctis E, F, G; </s>
  <s xml:id="echoid-s14609" xml:space="preserve">per hæc verò ducantur _Tangentes_ BQ, ER, FS, GT <lb/>parallelis ZE, ZF, ZG, DA occurrentes punctis Q, R, S, T; <lb/></s>
  <s xml:id="echoid-s14610" xml:space="preserve">habebit recta ADad omnes interceptas EQ, FR, GS, AT(ſi-<lb/>mul ſumptas) rationem quàvis aſſignabili majorem.</s>
  <s xml:id="echoid-s14611" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14612" xml:space="preserve">Nam ducantur rectæ EY, FX, GV ad BD parallelæ. </s>
  <s xml:id="echoid-s14613" xml:space="preserve">Habent <lb/>igitur rectæ ZE, YF, XG, VA ad rectas EQ, FR, GS, AT (ſin-<lb/>
<anchor type="note" xlink:label="note-0294-02a" xlink:href="note-0294-02"/>
gulæ ad ſingulas ſibi in directum poſitas reſpectivè) rationem deſigna-<lb/>bili quâcunque majorem. </s>
  <s xml:id="echoid-s14614" xml:space="preserve">ergò ſimul omnes iſtæ ad has ſimul omnes <lb/>_rationem_ habent deſignabili quâvis _majorem;_ </s>
  <s xml:id="echoid-s14615" xml:space="preserve">hoc eſt recta AD ad EQ <lb/>+ FR + GS + AT ejuſmodi rationem habet.</s>
  <s xml:id="echoid-s14616" xml:space="preserve"/>
</p>
<div xml:id="echoid-div476" type="float" level="2" n="2">
<note position="left" xlink:label="note-0294-02" xlink:href="note-0294-02a" xml:space="preserve">Fig. 175.</note>
</div>
<p>
  <s xml:id="echoid-s14617" xml:space="preserve">V. </s>
  <s xml:id="echoid-s14618" xml:space="preserve">Hinc inter computandum, omnes EQ, FR, GS, AT ſimul ac-<lb/>ceptæ nihilo æquivalent; </s>
  <s xml:id="echoid-s14619" xml:space="preserve">ſeu rectæ ZE, ZQ; </s>
  <s xml:id="echoid-s14620" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14621" xml:space="preserve">ZF, YR, &amp;</s>
  <s xml:id="echoid-s14622" xml:space="preserve">c. </s>
  <s xml:id="echoid-s14623" xml:space="preserve">æ-<lb/>quantur; </s>
  <s xml:id="echoid-s14624" xml:space="preserve">item tangentium particulæ BQ, ER, &amp;</s>
  <s xml:id="echoid-s14625" xml:space="preserve">c. </s>
  <s xml:id="echoid-s14626" xml:space="preserve">reſpectivis _curvœ_ <lb/>portiunculis BE, EF, &amp;</s>
  <s xml:id="echoid-s14627" xml:space="preserve">c. </s>
  <s xml:id="echoid-s14628" xml:space="preserve">pares, &amp; </s>
  <s xml:id="echoid-s14629" xml:space="preserve">quaſi coincidentes haberi poſſunt. <lb/></s>
  <s xml:id="echoid-s14630" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s14631" xml:space="preserve">adſumere tutò licet, quæ evidentèr his cohærent.</s>
  <s xml:id="echoid-s14632" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14633" xml:space="preserve">VI. </s>
  <s xml:id="echoid-s14634" xml:space="preserve">Sit porrò _curva_ quævis AB, cujus _Axis_ AD, &amp; </s>
  <s xml:id="echoid-s14635" xml:space="preserve">ad hunc <lb/>
<anchor type="note" xlink:label="note-0294-03a" xlink:href="note-0294-03"/>
applicata DB; </s>
  <s xml:id="echoid-s14636" xml:space="preserve">æquiſecetur autem DB in partes indefinitè multas ad <lb/>puncta Z, per quæ ducantur rectæ ad AD parallelæ, curvam AB <lb/>interſecantes punctis X; </s>
  <s xml:id="echoid-s14637" xml:space="preserve">quibus occurrant per ipſa X ductæ ad BD <lb/>parallelæ rectæ ME, NF, OG, PH; </s>
  <s xml:id="echoid-s14638" xml:space="preserve">ſit autem ſegmento ADB <lb/>(rectis AD, DB, &amp; </s>
  <s xml:id="echoid-s14639" xml:space="preserve">curvâ AB comprehenſo) _circumſcripta ſigura_ <lb/>ADBMXNXOXPXRA major _ſpatio_ quodam S; </s>
  <s xml:id="echoid-s14640" xml:space="preserve">dico _ſegmentum_ <lb/>ADB non eſſe minus quàm S.</s>
  <s xml:id="echoid-s14641" xml:space="preserve"/>
</p>
<div xml:id="echoid-div477" type="float" level="2" n="3">
<note position="left" xlink:label="note-0294-03" xlink:href="note-0294-03a" xml:space="preserve">Fig. 176.</note>
</div>
<p>
  <s xml:id="echoid-s14642" xml:space="preserve">Nam ſi ſieripoteſt ſit ADB minus quàm S exceſſu _rectangulaum_ <lb/>ADLKadæquante, &amp; </s>
  <s xml:id="echoid-s14643" xml:space="preserve">quoniam AReſt indefinitè parva, adeóque <lb/>minor quàm AK, liquet rectangulum ADZRminus eſſe _rectangulo_
<pb o="117" file="0295" n="310" rhead=""/>
ADLK. </s>
  <s xml:id="echoid-s14644" xml:space="preserve">item patet _ſegmentum_ ADB unà cum _rectangulo_ ADZR <lb/>majus eſſe _figurâ circumſcriptâ_ (etenim _rectangulum_ ADZR_rectan-_ <lb/>_gulis_ RH, PG, OF, NE, MZ æ quatur, proindéque majus eſt _irili-_ <lb/>_neis_ AXR, XXP, XXO, XXN, XBM). </s>
  <s xml:id="echoid-s14645" xml:space="preserve">ergò _ſegmentum_ ADB <lb/>
<anchor type="note" xlink:label="note-0295-01a" xlink:href="note-0295-01"/>
unà cum _rectangulo_ ADLK multo majus eſt _figurâ circumſcriptâ_; <lb/></s>
  <s xml:id="echoid-s14646" xml:space="preserve">hoc eſt, _ſpatium_ S majus eſt _figurâ circumſcriptâ_, contra _Hypotbeſin._</s>
  <s xml:id="echoid-s14647" xml:space="preserve"/>
</p>
<div xml:id="echoid-div478" type="float" level="2" n="4">
<note position="right" xlink:label="note-0295-01" xlink:href="note-0295-01a" xml:space="preserve">Fig. 176,</note>
</div>
<p>
  <s xml:id="echoid-s14648" xml:space="preserve">VII. </s>
  <s xml:id="echoid-s14649" xml:space="preserve">Item, ſi ponatur _figura inſcripta_ HXGXFXEXZDH minor <lb/>_ſpatio quodam_ S; </s>
  <s xml:id="echoid-s14650" xml:space="preserve">dico _ſegmentum_ ADB non eſſe majus quàm S.</s>
  <s xml:id="echoid-s14651" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14652" xml:space="preserve">Nam ſi majus eſſe velis, eſto rurſum _exceſſus par rectangulo_ ADLK; <lb/></s>
  <s xml:id="echoid-s14653" xml:space="preserve">quod utique (ſicut prius) majus erit _rectangulo_ ADZR. </s>
  <s xml:id="echoid-s14654" xml:space="preserve">Eſt autem <lb/>_ſegmentum_ ADB, dempto _rectangulo_ ADZR, minus figurâ inſcriptâ. </s>
  <s xml:id="echoid-s14655" xml:space="preserve"><lb/>ergò ſegmentum ADB, dempro rectangulo ADLK, multo minus fit <lb/>inſcriptâ; </s>
  <s xml:id="echoid-s14656" xml:space="preserve">hoc eſt _ſpatium_ S minus eſt inſcriptâ figurâ, contra <lb/>_Hypotbeſin._</s>
  <s xml:id="echoid-s14657" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14658" xml:space="preserve">VIII. </s>
  <s xml:id="echoid-s14659" xml:space="preserve">Hinc, ſi _ſp@tium_ quodcunque fuerit, (puta S) cui circumſcripta <lb/>figura æquetur _ſigurœ_ ADBMNOPRA; </s>
  <s xml:id="echoid-s14660" xml:space="preserve">nec non cui _inſcripta figura_ <lb/>_æquetur figuræ_ HGFEZDH; </s>
  <s xml:id="echoid-s14661" xml:space="preserve">palàm eſt _ſpatium_ iſtud S _ſegmento_ <lb/>ADB exæquari.</s>
  <s xml:id="echoid-s14662" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14663" xml:space="preserve">Nam (utì mox oſtenſum) hoc illo majus eſſe nequit, aut mi-<lb/>nus.</s>
  <s xml:id="echoid-s14664" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14665" xml:space="preserve">Poterunt autem hæc ad _alios circumſcriptionis ac inſcriptionis modos_ <lb/>accomodari. </s>
  <s xml:id="echoid-s14666" xml:space="preserve">ſuffecerit innuiſſe.</s>
  <s xml:id="echoid-s14667" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div480" type="section" level="1" n="47">
<head xml:id="echoid-head50" style="it" xml:space="preserve">Conicorum Superſicies dimetiendi Metbodus.</head>
<p>
  <s xml:id="echoid-s14668" xml:space="preserve">S It _curva_ quæpiam AMB, cujus _Axis_ AD, &amp; </s>
  <s xml:id="echoid-s14669" xml:space="preserve">in hoc ſignatum <lb/>
<anchor type="note" xlink:label="note-0295-02a" xlink:href="note-0295-02"/>
punctum C; </s>
  <s xml:id="echoid-s14670" xml:space="preserve">ad ipſum vero ordinata recta BD. </s>
  <s xml:id="echoid-s14671" xml:space="preserve">à puncto quo-<lb/>piam M in curva ſumpto ducatur recta ME curvam tangens, &amp; </s>
  <s xml:id="echoid-s14672" xml:space="preserve">à C <lb/>demittatur CGad ME perpendicularis; </s>
  <s xml:id="echoid-s14673" xml:space="preserve">ſit item determinata recta <lb/>CV ad planam DAB recta, &amp; </s>
  <s xml:id="echoid-s14674" xml:space="preserve">connectatur VG(erit VG _ipſi_ MG <lb/>perpendicularis; </s>
  <s xml:id="echoid-s14675" xml:space="preserve">nam ſi ducatur CHad GM parallela, liquet CH <lb/>plano GVG rectam eſſe, adeoque GMeidem recta erit) Porrò ſit <lb/>linea RStalis, ut ductâ rectâ MIX ad AD parallelâ (quæ ſecet or-
<pb o="118" file="0296" n="311" rhead=""/>
dinatam BDin I, &amp; </s>
  <s xml:id="echoid-s14676" xml:space="preserve">lineam RSin X) ſit MP. </s>
  <s xml:id="echoid-s14677" xml:space="preserve">ME:</s>
  <s xml:id="echoid-s14678" xml:space="preserve">: VG. </s>
  <s xml:id="echoid-s14679" xml:space="preserve">IX; <lb/></s>
  <s xml:id="echoid-s14680" xml:space="preserve">vel, ſit linea AL talis, ut ductâ MPY ad BDparallelâ (quæ ſecet <lb/>axem ADin P, &amp; </s>
  <s xml:id="echoid-s14681" xml:space="preserve">lineam ALin Y) ſit PE. </s>
  <s xml:id="echoid-s14682" xml:space="preserve">ME:</s>
  <s xml:id="echoid-s14683" xml:space="preserve">: VG. </s>
  <s xml:id="echoid-s14684" xml:space="preserve">PY; </s>
  <s xml:id="echoid-s14685" xml:space="preserve">erit <lb/>
<anchor type="note" xlink:label="note-0296-01a" xlink:href="note-0296-01"/>
tunc utrumque _ſpatium_ (ſingillatim) BRS D, vel ADL duplum _ſu-_ <lb/>_perfici@i conicœ_, quod ex recta per V &amp; </s>
  <s xml:id="echoid-s14686" xml:space="preserve">curvam AMB mota progene-<lb/>ratur.</s>
  <s xml:id="echoid-s14687" xml:space="preserve"/>
</p>
<div xml:id="echoid-div480" type="float" level="2" n="1">
<note position="right" xlink:label="note-0295-02" xlink:href="note-0295-02a" xml:space="preserve">Fig. 177.</note>
<note position="left" xlink:label="note-0296-01" xlink:href="note-0296-01a" xml:space="preserve">Fig. 177.</note>
</div>
<p>
  <s xml:id="echoid-s14688" xml:space="preserve">Nam ſumatur MNindefinita curvæ particula; </s>
  <s xml:id="echoid-s14689" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14690" xml:space="preserve">per N ducantur <lb/>rectæ NOKTad ipſam AD, &amp; </s>
  <s xml:id="echoid-s14691" xml:space="preserve">NQZ ad BDparallelæ (quæ li-<lb/>neas expoſitas, ut _Schema_ monſtrat, ſecent) connectantúrque rectæ <lb/>VM, VN. </s>
  <s xml:id="echoid-s14692" xml:space="preserve">eſtque MO. </s>
  <s xml:id="echoid-s14693" xml:space="preserve">MN:</s>
  <s xml:id="echoid-s14694" xml:space="preserve">: MP. </s>
  <s xml:id="echoid-s14695" xml:space="preserve">MF:</s>
  <s xml:id="echoid-s14696" xml:space="preserve">: VG. </s>
  <s xml:id="echoid-s14697" xml:space="preserve">IX. </s>
  <s xml:id="echoid-s14698" xml:space="preserve">quare <lb/>MN x VG = MO x IX = IK x IX. </s>
  <s xml:id="echoid-s14699" xml:space="preserve">Item eſt NO. </s>
  <s xml:id="echoid-s14700" xml:space="preserve">MN:</s>
  <s xml:id="echoid-s14701" xml:space="preserve">: PE. <lb/></s>
  <s xml:id="echoid-s14702" xml:space="preserve">ME:</s>
  <s xml:id="echoid-s14703" xml:space="preserve">: VG. </s>
  <s xml:id="echoid-s14704" xml:space="preserve">PY. </s>
  <s xml:id="echoid-s14705" xml:space="preserve">unde MN x VG = NO x PY = QP x PY. </s>
  <s xml:id="echoid-s14706" xml:space="preserve"><lb/>Eſt autem MN x VG duplum trianguli MVN. </s>
  <s xml:id="echoid-s14707" xml:space="preserve">quapropter tam IK <lb/>x IX, quàm QP x PY duplum eſt _trianguli_ MVN. </s>
  <s xml:id="echoid-s14708" xml:space="preserve">pariter autem <lb/>ubique fit. </s>
  <s xml:id="echoid-s14709" xml:space="preserve">ergò conſtat Propoſitum.</s>
  <s xml:id="echoid-s14710" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div482" type="section" level="1" n="48">
<head xml:id="echoid-head51" style="it" xml:space="preserve">Exemplum.</head>
<p>
  <s xml:id="echoid-s14711" xml:space="preserve">Sit curva AMB _byperbola æquilatera_, cujus _Centrum_ C, ſitque <lb/>
<anchor type="note" xlink:label="note-0296-02a" xlink:href="note-0296-02"/>
CV = CA = _r._ </s>
  <s xml:id="echoid-s14712" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14713" xml:space="preserve">CP = _x_ (nam hujuſmodi _calculo_ plerunque <lb/>rem expedit peragere) tum connexâ MC; </s>
  <s xml:id="echoid-s14714" xml:space="preserve">patet eſſe EC = {_rr_/_x_}; <lb/></s>
  <s xml:id="echoid-s14715" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14716" xml:space="preserve">MCq = 2 _xx_ - _rr_ (nam PMq = _xx_ - _rr_) item eſt MCq. </s>
  <s xml:id="echoid-s14717" xml:space="preserve"><lb/>CPq:</s>
  <s xml:id="echoid-s14718" xml:space="preserve">: MEq. </s>
  <s xml:id="echoid-s14719" xml:space="preserve">MPq; </s>
  <s xml:id="echoid-s14720" xml:space="preserve">hoc eſt MCq.</s>
  <s xml:id="echoid-s14721" xml:space="preserve">CPq:</s>
  <s xml:id="echoid-s14722" xml:space="preserve">: ECq. </s>
  <s xml:id="echoid-s14723" xml:space="preserve">CGq. </s>
  <s xml:id="echoid-s14724" xml:space="preserve">hoc <lb/>eſt 2 _xx_ - _rr_. </s>
  <s xml:id="echoid-s14725" xml:space="preserve">_xx_:</s>
  <s xml:id="echoid-s14726" xml:space="preserve">: {_r_<emph style="sub">4</emph>/_xx_}. </s>
  <s xml:id="echoid-s14727" xml:space="preserve">CGq = {_r_<emph style="sub">4</emph>/2 _xx_ - _rr_}. </s>
  <s xml:id="echoid-s14728" xml:space="preserve">quare VGq = {_r_<emph style="sub">4</emph>/2 _xx_ - _rr_} + <lb/>_rr_ = {2 _rrxx_/2 _xx_ - _rr_} = {VAq x CPq/MCq}.</s>
  <s xml:id="echoid-s14729" xml:space="preserve">vel VG = {VA x CP/MC}. </s>
  <s xml:id="echoid-s14730" xml:space="preserve">quare <lb/>VG. </s>
  <s xml:id="echoid-s14731" xml:space="preserve">VA:</s>
  <s xml:id="echoid-s14732" xml:space="preserve">: (CP. </s>
  <s xml:id="echoid-s14733" xml:space="preserve">MC):</s>
  <s xml:id="echoid-s14734" xml:space="preserve">: MP. </s>
  <s xml:id="echoid-s14735" xml:space="preserve">ME. </s>
  <s xml:id="echoid-s14736" xml:space="preserve">hinc conſectatur in hoc <lb/>caſu, quum ubique ſit IX = VA, _lineam_ RS fore _rectam_; </s>
  <s xml:id="echoid-s14737" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14738" xml:space="preserve">_rectan-_ <lb/>_gulum_ BRSD _ſuperficiei conicœ_ AMBV _duplum eſſe._</s>
  <s xml:id="echoid-s14739" xml:space="preserve"/>
</p>
<div xml:id="echoid-div482" type="float" level="2" n="1">
<note position="left" xlink:label="note-0296-02" xlink:href="note-0296-02a" xml:space="preserve">Fig. 177.</note>
</div>
<p>
  <s xml:id="echoid-s14740" xml:space="preserve">Cæterùm hoc _elegans exemplum_ ſuppeditavit Generoſus, ingenio ac <lb/>eruditione præſtans, Vir (_Collegii noſtri, quod olim Sociorum Com-_ <lb/>_menſalis incoluit_, ornamentum) D. </s>
  <s xml:id="echoid-s14741" xml:space="preserve">_Franciſcus Feſſopius_, Armiger; <lb/></s>
  <s xml:id="echoid-s14742" xml:space="preserve">cujus in hanc rem perquam ingenioſo mihi comiter impertito ſcripto <lb/>(ipſius injuſſu quidem, at ſpero non ingratiis) ſeu _Gemmâ_ quâdam au-<lb/>debo mea condecorare.</s>
  <s xml:id="echoid-s14743" xml:space="preserve"/>
</p>
<pb o="119" file="0297" n="312" rhead=""/>
</div>
<div xml:id="echoid-div484" type="section" level="1" n="49">
<head xml:id="echoid-head52" style="it" xml:space="preserve">Prop. 1.</head>
<p>
  <s xml:id="echoid-s14744" xml:space="preserve">Si à puncto E in _axe A m coni recti_ ABC _p_ recta infinita EC <lb/>tranſeat per _coni ſuperficiem_, &amp; </s>
  <s xml:id="echoid-s14745" xml:space="preserve">quieſcente termino E circumferatur <lb/>
<anchor type="note" xlink:label="note-0297-01a" xlink:href="note-0297-01"/>
recta ECdonec redeat ad locum à quo coepit moveri, ita ut femper <lb/>aliqua pars ejus ſecet _coni ſuperficiem_ (puta per H) _perbolam_ CFD &amp; </s>
  <s xml:id="echoid-s14746" xml:space="preserve"><lb/>rectas DAA Cin ſuperficie coni ſitas) _ſolidum comprebenſum à ſuper-_ <lb/>_ficie vel ſuperficiebus genitis à linea_ EC ſic mota &amp; </s>
  <s xml:id="echoid-s14747" xml:space="preserve">à _portione ſuperft-_ <lb/>_ciei_ ejuſdem coni terminatæ à linea vel lineis CFD, DA, ACquas <lb/>recta ECcircumlata deſcribit in _ſuperficie conica_, erit æquale _Pyra-_ <lb/>_midi_ cujus _Altitudo_ eſt æ qualis _perpendiculari_ E _n_ à puncto E ad latus <lb/>_Coni_ deductæ _b@ſis_ verò æqualis eidem _ſuperficiei conicœ terminat œ à_ <lb/>linea vel lineis CFD, DA, ACgeneratis à motu lineæ EC.</s>
  <s xml:id="echoid-s14748" xml:space="preserve"/>
</p>
<div xml:id="echoid-div484" type="float" level="2" n="1">
<note position="right" xlink:label="note-0297-01" xlink:href="note-0297-01a" xml:space="preserve">Fig. 178.</note>
</div>
<p>
  <s xml:id="echoid-s14749" xml:space="preserve">_Solidum_ enim ECF, DAC conſtat ex _infinitis pyramidibus_ EC _o_ A <lb/>E _o o_ A, &amp;</s>
  <s xml:id="echoid-s14750" xml:space="preserve">c. </s>
  <s xml:id="echoid-s14751" xml:space="preserve">æquialtis perpendiculari E n, quarum baſes omnes <lb/>ſimul ſumptæ, exhauriunt _ſuperficiem conicam_ CFD, DA, AC.</s>
  <s xml:id="echoid-s14752" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div486" type="section" level="1" n="50">
<head xml:id="echoid-head53" style="it" xml:space="preserve">Prop. 2.</head>
<p>
  <s xml:id="echoid-s14753" xml:space="preserve">Datus ſit _Conus rectus_ ABC _p_ ſecetur à plano CFD axi A _m_ pa-<lb/>
<anchor type="note" xlink:label="note-0297-02a" xlink:href="note-0297-02"/>
rallelo ducantur rectæ AC, ADà vertice _coni_ ad _lineam byperbolicam_ <lb/>CFD, &amp; </s>
  <s xml:id="echoid-s14754" xml:space="preserve">ſuper _triangulo_ ACD erigatur _pyramis_ EACD habens <lb/>_verticem_ E in _axe coni_; </s>
  <s xml:id="echoid-s14755" xml:space="preserve">ſitque E δ plano ACD perpendicularis, &amp; </s>
  <s xml:id="echoid-s14756" xml:space="preserve"><lb/>E _n_ lateri coni.</s>
  <s xml:id="echoid-s14757" xml:space="preserve"/>
</p>
<div xml:id="echoid-div486" type="float" level="2" n="1">
<note position="right" xlink:label="note-0297-02" xlink:href="note-0297-02a" xml:space="preserve">Fig. 178.</note>
</div>
<p>
  <s xml:id="echoid-s14758" xml:space="preserve">Dico, _ſuperficies conica_ terminata à _linea byperbolica_ CFD &amp; </s>
  <s xml:id="echoid-s14759" xml:space="preserve">re-<lb/>ctis DA, ACita ſe habet ad ACD _baſem pyramidis_ EACD ut <lb/>_altitudo_ E δ _pyramidis_ EACD ad perpendiculum E _n._ </s>
  <s xml:id="echoid-s14760" xml:space="preserve">Quoniam <lb/>enim Conici ACF D, ECFD habent vertices A &amp; </s>
  <s xml:id="echoid-s14761" xml:space="preserve">E in plano baſi <lb/>CFD (quæ eſt utrique Conico communis) parallelo ergo ſunt æ-<lb/>quales. </s>
  <s xml:id="echoid-s14762" xml:space="preserve">Si ergò à ſolido quod componitur à conico ACFDaddito <lb/>pyramide ECADauferatur conicus ECFDreliquum erit ſolidum <lb/>ECFDACquale in propoſitione prima deſcribitur motu rectæ EC <lb/>æquale pyramidi EAC D. </s>
  <s xml:id="echoid-s14763" xml:space="preserve">Quoniam verò _œqualium pyramidum_ re-<lb/>ciprocæ ſunt _baſes al@itudinibus_, ut _altitudo_ E δ _pyramidis_ EACD <lb/>ad perpendiculum E _n_ ita erit _ſuperſicies conica_ terminata à _linea by-_ <lb/>_perbolica_ CFD &amp; </s>
  <s xml:id="echoid-s14764" xml:space="preserve">rectis DA, ACad Triangulum ACD. </s>
  <s xml:id="echoid-s14765" xml:space="preserve">q. </s>
  <s xml:id="echoid-s14766" xml:space="preserve">E. </s>
  <s xml:id="echoid-s14767" xml:space="preserve">D.</s>
  <s xml:id="echoid-s14768" xml:space="preserve"/>
</p>
<pb o="120" file="0298" n="313" rhead=""/>
</div>
<div xml:id="echoid-div488" type="section" level="1" n="51">
<head xml:id="echoid-head54" xml:space="preserve">Prop. 3.</head>
<p>
  <s xml:id="echoid-s14769" xml:space="preserve">Datus ſit _Conus rectus_ ABC _p._ </s>
  <s xml:id="echoid-s14770" xml:space="preserve">Secetur à plano (puta _triangulo_ <lb/>
<anchor type="note" xlink:label="note-0298-01a" xlink:href="note-0298-01"/>
_qrt_) quod quidem planum ſecabit _axem coni_ in puncto _q_ ſupra _verti-_ <lb/>_cem_ productum &amp; </s>
  <s xml:id="echoid-s14771" xml:space="preserve">in communi interſectione cum _ſuperficie coni_ habe-<lb/>bit _lineam byperbolicam_ RS_t_ ducantur à vertice coni A rectæ A _r_, A _t_, <lb/>à puncto _q_ demittatur perpendiculum _q_ X lateri coni A _p_ producto &amp; </s>
  <s xml:id="echoid-s14772" xml:space="preserve">à <lb/>puncto A perpendiculum AZplano _qrt._</s>
  <s xml:id="echoid-s14773" xml:space="preserve"/>
</p>
<div xml:id="echoid-div488" type="float" level="2" n="1">
<note position="left" xlink:label="note-0298-01" xlink:href="note-0298-01a" xml:space="preserve">Fig. 178.</note>
</div>
<p>
  <s xml:id="echoid-s14774" xml:space="preserve">Dico _ſuperficies contca_ terminata à _linca byperbolica, rst_ &amp; </s>
  <s xml:id="echoid-s14775" xml:space="preserve">rectis <lb/>_r_ A, _t_ A, ita ſe habet ad _figuram byperbolicam cavam qrstq_ ut _perpen-_ <lb/>_diculum_ AZad _perpendiculum q_ X.</s>
  <s xml:id="echoid-s14776" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14777" xml:space="preserve">Recta enim _qr_, circumlata, quieſcente termino _q_ per lineas _rst, t_ A, Ar <lb/>generat tres _ſuperficies_, nempe _byperbolicam cavam qr, st_, &amp; </s>
  <s xml:id="echoid-s14778" xml:space="preserve">_duo tri-_ <lb/>_angula qt_ A, _q_ A _r_, quæ unà cum _ſuperficie conica_ terminata à lineis <lb/>_rst, t_ A, A _r_, comprehendunt _Solidum qrs, t_ A _r._ </s>
  <s xml:id="echoid-s14779" xml:space="preserve">Hoc verò _ſolidum_ <lb/>_œguale_ eſt _pyramidi_ cujus _altitudo_ eſt æqualis perpendiculo _q_ X, nam <lb/>infinitæ pyramides _q_ A _r_ V, _q_ AVV, exhauriunt ſolidum _qr_ S _t_ A _r._ <lb/></s>
  <s xml:id="echoid-s14780" xml:space="preserve">Si verò aliter contemplari volumus, hoc ſolidum _qrst_ A _r_ poteſt con-<lb/>ſideraritanquam _ſigura @onica_ A _r_ S _tqr_ habens pro _baſe figuram by-_ <lb/>_perbolicam_ cavam _qr_ S _tq_, &amp; </s>
  <s xml:id="echoid-s14781" xml:space="preserve">pro altitudine _perpendiculum_ AZ. </s>
  <s xml:id="echoid-s14782" xml:space="preserve">Ergò <lb/>reciprocando _baſes altitudinibus_, ut AZad q X, ita _ſuperficies, r_ S t A _r_ <lb/>ad _figuram byperbolicam cavam qr_ S _tq._</s>
  <s xml:id="echoid-s14783" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div490" type="section" level="1" n="52">
<head xml:id="echoid-head55" xml:space="preserve">Prop. 4.</head>
<p>
  <s xml:id="echoid-s14784" xml:space="preserve">Datus ſit _Conus rectus_ AB _b g_ ſecetur à plano HFEGper axem <lb/>infra verticem, a puncto H ubi _planum_ fecat _axem coni_, demittatur HK <lb/>
<anchor type="note" xlink:label="note-0298-02a" xlink:href="note-0298-02"/>
_perpendiculum_ lateri cuilibet coni &amp; </s>
  <s xml:id="echoid-s14785" xml:space="preserve">à verticè A _perpendiculum_ ALpla-<lb/>no HFE G.</s>
  <s xml:id="echoid-s14786" xml:space="preserve"/>
</p>
<div xml:id="echoid-div490" type="float" level="2" n="1">
<note position="left" xlink:label="note-0298-02" xlink:href="note-0298-02a" xml:space="preserve">Fig. 179.</note>
</div>
<p>
  <s xml:id="echoid-s14787" xml:space="preserve">Dico, _Superſicies conica_ terminata a lineis FECGAAF ita ſe <lb/>habebit ad _planum_ HFEG ut _perpendiculum_ AL ad _perpendiculum_ <lb/>H K.</s>
  <s xml:id="echoid-s14788" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14789" xml:space="preserve">Probatur eodem fere eodem fere argumento quo ſuperior.</s>
  <s xml:id="echoid-s14790" xml:space="preserve"/>
</p>
<pb o="121" file="0299" n="314" rhead=""/>
</div>
<div xml:id="echoid-div492" type="section" level="1" n="53">
<head xml:id="echoid-head56" xml:space="preserve">APPENDICULA 3.</head>
<p>
  <s xml:id="echoid-s14791" xml:space="preserve">Præcedentia recolenti nonnulla videntur elapſa; </s>
  <s xml:id="echoid-s14792" xml:space="preserve">quæ forſan ex uſu <lb/>ſit adjicere. </s>
  <s xml:id="echoid-s14793" xml:space="preserve">_Demònſtrationes_ elicere poterit quiſpiam è præmiſſis; </s>
  <s xml:id="echoid-s14794" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14795" xml:space="preserve"><lb/>potior inde fructus emerget.</s>
  <s xml:id="echoid-s14796" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div493" type="section" level="1" n="54">
<head xml:id="echoid-head57" xml:space="preserve">Problema I.</head>
<note position="right" xml:space="preserve">Fig. 180.</note>
<p>
  <s xml:id="echoid-s14797" xml:space="preserve">Sit _curva_ quævis KEG, cujus _axis_ AD; </s>
  <s xml:id="echoid-s14798" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14799" xml:space="preserve">in hoc ſignatum <lb/>punctum A; </s>
  <s xml:id="echoid-s14800" xml:space="preserve">curva reperiatur, puta LMB, talis, ut ſi ductâ utcun-<lb/>que rectâ PEM axi ADperpendicularis curvam KEG ſecet in E, &amp; </s>
  <s xml:id="echoid-s14801" xml:space="preserve"><lb/>curvam LMB in M; </s>
  <s xml:id="echoid-s14802" xml:space="preserve">nec non connectatur AE, &amp; </s>
  <s xml:id="echoid-s14803" xml:space="preserve">curvam LMB <lb/>tangat recta TM; </s>
  <s xml:id="echoid-s14804" xml:space="preserve">ſit TMipſi AEparallela.</s>
  <s xml:id="echoid-s14805" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14806" xml:space="preserve">Hoc ità fiet. </s>
  <s xml:id="echoid-s14807" xml:space="preserve">Per aliquodcunque punctum R, in axe AD fumptum, <lb/>protendatur recta RZad ipſam ADperpendicularis; </s>
  <s xml:id="echoid-s14808" xml:space="preserve">cui occurrat re-<lb/>cta EAproducta in S; </s>
  <s xml:id="echoid-s14809" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14810" xml:space="preserve">in recta EPſumatur PY = RS; </s>
  <s xml:id="echoid-s14811" xml:space="preserve">ità de-<lb/>terminetur curvæ OYY proprietas; </s>
  <s xml:id="echoid-s14812" xml:space="preserve">tum ſit rectangulum ex AR, &amp; </s>
  <s xml:id="echoid-s14813" xml:space="preserve"><lb/>PMæquale ſpatio AYYP(ſeu PM = {ſpat AYYP/AR}) habebit <lb/>curva LMMBconditionem propoſitam.</s>
  <s xml:id="echoid-s14814" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14815" xml:space="preserve">Adnotari poteft, ſi ſtantibus reliquis, ſit curva QXX talis, ut cum <lb/>hanc ſecet recta E Pin X, ſit PX = AS; </s>
  <s xml:id="echoid-s14816" xml:space="preserve">erit ſpatium AXXP <lb/>æqualerectangulo ex AR, &amp; </s>
  <s xml:id="echoid-s14817" xml:space="preserve">curva LM, ſeu {AXXP/AR} = LM.</s>
  <s xml:id="echoid-s14818" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div494" type="section" level="1" n="55">
<head xml:id="echoid-head58" xml:space="preserve">Exemp. I.</head>
<p>
  <s xml:id="echoid-s14819" xml:space="preserve">Sit ADG _circuli_ quadrans, &amp; </s>
  <s xml:id="echoid-s14820" xml:space="preserve">ductâ EPad ADutcunque per-<lb/>pendiculari, connexâque DE; </s>
  <s xml:id="echoid-s14821" xml:space="preserve">deſignetur curva AMB talis, ut ſi <lb/>
<anchor type="note" xlink:label="note-0299-02a" xlink:href="note-0299-02"/>
producta recta EPM hanc ſecet in M, ipſamque tangat recta MT, <lb/>ſit MTad DEparallela. </s>
  <s xml:id="echoid-s14822" xml:space="preserve">Hocita peragetur. </s>
  <s xml:id="echoid-s14823" xml:space="preserve">Ducatur AZad DG <lb/>parallela; </s>
  <s xml:id="echoid-s14824" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14825" xml:space="preserve">huic occurrat producta DEin S, &amp; </s>
  <s xml:id="echoid-s14826" xml:space="preserve">curva AYY talis <lb/>ſit, ut ſi hanc ſecet producta PEin Y, ſit PY = AS; </s>
  <s xml:id="echoid-s14827" xml:space="preserve">tum capiatur <lb/>PM = {Spat. </s>
  <s xml:id="echoid-s14828" xml:space="preserve">AYP/AD}; </s>
  <s xml:id="echoid-s14829" xml:space="preserve">factum erit.</s>
  <s xml:id="echoid-s14830" xml:space="preserve"/>
</p>
<div xml:id="echoid-div494" type="float" level="2" n="1">
<note position="right" xlink:label="note-0299-02" xlink:href="note-0299-02a" xml:space="preserve">Fig. 181.</note>
</div>
<p>
  <s xml:id="echoid-s14831" xml:space="preserve">Not. </s>
  <s xml:id="echoid-s14832" xml:space="preserve">Quòd ſi curva QXX talis ſit, ut PX = DS (vel ſi AQ <lb/> = AD, &amp; </s>
  <s xml:id="echoid-s14833" xml:space="preserve">QXX ſit _byperbola_ angulo ADG comprehenſa) erit <lb/>curva AM x AD = ſpat. </s>
  <s xml:id="echoid-s14834" xml:space="preserve">AQX P.</s>
  <s xml:id="echoid-s14835" xml:space="preserve"/>
</p>
<pb o="122" file="0300" n="315" rhead=""/>
</div>
<div xml:id="echoid-div496" type="section" level="1" n="56">
<head xml:id="echoid-head59" style="it" xml:space="preserve">Exemp. II.</head>
<p>
  <s xml:id="echoid-s14836" xml:space="preserve">Sit curva AEG (cnjus Axis AD) proprietate talis, ut ſi à quo-<lb/>cunque puncto in ipſa ſumpto E, ducatur recta EPad AD normalis; <lb/></s>
  <s xml:id="echoid-s14837" xml:space="preserve">
<anchor type="note" xlink:label="note-0300-01a" xlink:href="note-0300-01"/>
connectatúrque AE, ſit AEinter deſignatam AR, &amp; </s>
  <s xml:id="echoid-s14838" xml:space="preserve">APpropor-<lb/>tione media, ſecundum ordinem, cujus exponens ſit {_n_/_m_}; </s>
  <s xml:id="echoid-s14839" xml:space="preserve">reperiatur <lb/>curva AMB, quam tangat TMad AEparallela.</s>
  <s xml:id="echoid-s14840" xml:space="preserve"/>
</p>
<div xml:id="echoid-div496" type="float" level="2" n="1">
<note position="left" xlink:label="note-0300-01" xlink:href="note-0300-01a" xml:space="preserve">Fig. 182.</note>
</div>
<p>
  <s xml:id="echoid-s14841" xml:space="preserve">De curva AMadnoto fore _n. </s>
  <s xml:id="echoid-s14842" xml:space="preserve">m_:</s>
  <s xml:id="echoid-s14843" xml:space="preserve">: AE. </s>
  <s xml:id="echoid-s14844" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s14845" xml:space="preserve">AM.</s>
  <s xml:id="echoid-s14846" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14847" xml:space="preserve">Si {_n_/_m_} = {1/2} (vel AEſit inter AR, AP ſimpliciter media) erit <lb/>AEG circulus, &amp; </s>
  <s xml:id="echoid-s14848" xml:space="preserve">AMB _Ciclois primaria_; </s>
  <s xml:id="echoid-s14849" xml:space="preserve">hujus igitur dimenſio è <lb/>lege generali habetur.</s>
  <s xml:id="echoid-s14850" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14851" xml:space="preserve">Hæc etiam ex adjuncto _Problemate_ magis ccomprehenſivo pera-<lb/>guntur.</s>
  <s xml:id="echoid-s14852" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div498" type="section" level="1" n="57">
<head xml:id="echoid-head60" style="it" xml:space="preserve">Probl. II.</head>
<p>
  <s xml:id="echoid-s14853" xml:space="preserve">Curva deſignetur, puta AMB, cujus _axis_ AD, ità ut in hac <lb/>
<anchor type="note" xlink:label="note-0300-02a" xlink:href="note-0300-02"/>
ſumpto puncto quopiam M, &amp; </s>
  <s xml:id="echoid-s14854" xml:space="preserve">ductâ MPad AD perpendiculâri, &amp; </s>
  <s xml:id="echoid-s14855" xml:space="preserve"><lb/>poſito rectam MT ipſam tangere, habeant TP, PM relationem aſ-<lb/>ſignatam.</s>
  <s xml:id="echoid-s14856" xml:space="preserve"/>
</p>
<div xml:id="echoid-div498" type="float" level="2" n="1">
<note position="left" xlink:label="note-0300-02" xlink:href="note-0300-02a" xml:space="preserve">Fig. 183.</note>
</div>
<p>
  <s xml:id="echoid-s14857" xml:space="preserve">Accipiatur recta quæpiam R, &amp; </s>
  <s xml:id="echoid-s14858" xml:space="preserve">fiat ut TPad PM (quam utique <lb/>rationem aſſignatâ dabit relatio) ità R ad PY (quæ nempe ſumatur <lb/>in recta PM, &amp; </s>
  <s xml:id="echoid-s14859" xml:space="preserve">ad axem ADordinetur) ſic ut per ejuſmodi puncta <lb/>Y tranſeat curva YYK; </s>
  <s xml:id="echoid-s14860" xml:space="preserve">tum ſi ſiat PM = {ſpat. </s>
  <s xml:id="echoid-s14861" xml:space="preserve">APY/R}; </s>
  <s xml:id="echoid-s14862" xml:space="preserve">de curvæ <lb/>AMB indè conſtabit natura.</s>
  <s xml:id="echoid-s14863" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div500" type="section" level="1" n="58">
<head xml:id="echoid-head61" style="it" xml:space="preserve">Exemp. I.</head>
<p>
  <s xml:id="echoid-s14864" xml:space="preserve">Sit ADG _circuli_ quadrans; </s>
  <s xml:id="echoid-s14865" xml:space="preserve">cujus radius æquetur deſignatæ R; </s>
  <s xml:id="echoid-s14866" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14867" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0300-03a" xlink:href="note-0300-03"/>
habere debeat TPad PM rationem eandem quam habet R ad arcum <lb/>AE; </s>
  <s xml:id="echoid-s14868" xml:space="preserve">ergo quum ſit, juxta præſcriptum, R. </s>
  <s xml:id="echoid-s14869" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s14870" xml:space="preserve">AE:</s>
  <s xml:id="echoid-s14871" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s14872" xml:space="preserve">PY; </s>
  <s xml:id="echoid-s14873" xml:space="preserve">e-<lb/>rit PY = arc. </s>
  <s xml:id="echoid-s14874" xml:space="preserve">AE; </s>
  <s xml:id="echoid-s14875" xml:space="preserve">hinc habetur PM = {APY/R}</s>
</p>
<div xml:id="echoid-div500" type="float" level="2" n="1">
<note position="left" xlink:label="note-0300-03" xlink:href="note-0300-03a" xml:space="preserve">Fig. 184.</note>
</div>
<pb o="123" file="0301" n="316" rhead=""/>
</div>
<div xml:id="echoid-div502" type="section" level="1" n="59">
<head xml:id="echoid-head62" xml:space="preserve">_Exemp_. II.</head>
<p>
  <s xml:id="echoid-s14876" xml:space="preserve">Sit ADG _circuli_ quâdrans, &amp; </s>
  <s xml:id="echoid-s14877" xml:space="preserve">habere debeat TP ad PM ratio-<lb/>nem eandem quam PE ad R; </s>
  <s xml:id="echoid-s14878" xml:space="preserve">eſt ergo PY æqualis _tangenti_ arcûs GE; <lb/></s>
  <s xml:id="echoid-s14879" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14880" xml:space="preserve">ſpat. </s>
  <s xml:id="echoid-s14881" xml:space="preserve">APYY = R x arc. </s>
  <s xml:id="echoid-s14882" xml:space="preserve">AE. </s>
  <s xml:id="echoid-s14883" xml:space="preserve">adeóque PM = arc. </s>
  <s xml:id="echoid-s14884" xml:space="preserve">AE.</s>
  <s xml:id="echoid-s14885" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div503" type="section" level="1" n="60">
<head xml:id="echoid-head63" xml:space="preserve">_Probl_. III.</head>
<p>
  <s xml:id="echoid-s14886" xml:space="preserve">Proponatur figura quælibet ADB (cujus _axis_ AD, _baſis_ DB) <lb/>
<anchor type="note" xlink:label="note-0301-01a" xlink:href="note-0301-01"/>
reperiatur curva KZL, proprietate talis, ut ductâ rectâ ZPM ad <lb/>DB utcunque parallela quæ lineas expoſitas ſecet ut cernis) poſitóque <lb/>rectam ZT tangere curvam KZL, ſit intercepta TP æqualis ipſi <lb/>PM.</s>
  <s xml:id="echoid-s14887" xml:space="preserve"/>
</p>
<div xml:id="echoid-div503" type="float" level="2" n="1">
<note position="right" xlink:label="note-0301-01" xlink:href="note-0301-01a" xml:space="preserve">Fig. 185.</note>
</div>
<p>
  <s xml:id="echoid-s14888" xml:space="preserve">Hocità perſicietur. </s>
  <s xml:id="echoid-s14889" xml:space="preserve">Sit curva OYY talis, ut adſumptâ quâdam <lb/>R, protractâque PMY, ſit PM. </s>
  <s xml:id="echoid-s14890" xml:space="preserve">R:</s>
  <s xml:id="echoid-s14891" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s14892" xml:space="preserve">PY; </s>
  <s xml:id="echoid-s14893" xml:space="preserve">tum liberè adſump-<lb/>tâ DL (in BD protensâ) ſit DL. </s>
  <s xml:id="echoid-s14894" xml:space="preserve">R:</s>
  <s xml:id="echoid-s14895" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s14896" xml:space="preserve">LE; </s>
  <s xml:id="echoid-s14897" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14898" xml:space="preserve">_aſymptotis_ DL, <lb/>DG per E deſcribatur _Hyperbola_ EXX; </s>
  <s xml:id="echoid-s14899" xml:space="preserve">tum ſit ſpatium LEXH æ-<lb/>quale ſpatio DOYP, &amp; </s>
  <s xml:id="echoid-s14900" xml:space="preserve">protractæ XH, YP concurrant in Z; </s>
  <s xml:id="echoid-s14901" xml:space="preserve">erit <lb/>Z in curva quæſita; </s>
  <s xml:id="echoid-s14902" xml:space="preserve">quam ſi tangat ZT, erit TP = PM.</s>
  <s xml:id="echoid-s14903" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14904" xml:space="preserve">Adnotetur, ſi propoſita ſigura ſit _rectangulum Parallelogrammum_ <lb/>ADBC, quod curvæ KZL hæc erit proprietas, ut ſit DH eodem <lb/>ordine inter DL, DO media _Geometricè_ proportionalis, quo DP <lb/>
<anchor type="note" xlink:label="note-0301-02a" xlink:href="note-0301-02"/>
inter DA &amp; </s>
  <s xml:id="echoid-s14905" xml:space="preserve">θ<unsure/> (ſeu nihilum) eſt media _Aritbmeticè_; </s>
  <s xml:id="echoid-s14906" xml:space="preserve">quod ſi liberè <lb/>juxta proprietatem hanc deſcribatur curva KZL, &amp; </s>
  <s xml:id="echoid-s14907" xml:space="preserve">_Mechanicè_ re-<lb/>periatur tangens ZT, indè quadrabitur _hyperbolicum ſpatium_ LEXH; <lb/></s>
  <s xml:id="echoid-s14908" xml:space="preserve">erit utique hoc æquale _rectangulo_ ex TP, AP.</s>
  <s xml:id="echoid-s14909" xml:space="preserve"/>
</p>
<div xml:id="echoid-div504" type="float" level="2" n="2">
<note position="right" xlink:label="note-0301-02" xlink:href="note-0301-02a" xml:space="preserve">Fig. 186.</note>
</div>
<p>
  <s xml:id="echoid-s14910" xml:space="preserve">Subnotari poſſit fore 1. </s>
  <s xml:id="echoid-s14911" xml:space="preserve">Spat. </s>
  <s xml:id="echoid-s14912" xml:space="preserve">ADLK = R x DL - DO. </s>
  <s xml:id="echoid-s14913" xml:space="preserve">2. </s>
  <s xml:id="echoid-s14914" xml:space="preserve">Sum. <lb/></s>
  <s xml:id="echoid-s14915" xml:space="preserve">mam ZPq = R x : </s>
  <s xml:id="echoid-s14916" xml:space="preserve">{DLq - DOq/2}. </s>
  <s xml:id="echoid-s14917" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14918" xml:space="preserve">ſummam ZP cub. </s>
  <s xml:id="echoid-s14919" xml:space="preserve">= R x <lb/>{DLcub. </s>
  <s xml:id="echoid-s14920" xml:space="preserve">- DOcub.</s>
  <s xml:id="echoid-s14921" xml:space="preserve">/3} &amp;</s>
  <s xml:id="echoid-s14922" xml:space="preserve">c<unsure/>. </s>
  <s xml:id="echoid-s14923" xml:space="preserve">3. </s>
  <s xml:id="echoid-s14924" xml:space="preserve">Siponatur φ eſſe centrum gr. </s>
  <s xml:id="echoid-s14925" xml:space="preserve">figu-<lb/>ræ ADLK, ducantúrque φψ ad AD, &amp; </s>
  <s xml:id="echoid-s14926" xml:space="preserve">φξ ad DL perpendicu-<lb/>lares, fore φψ = {DL + DO/4}, &amp; </s>
  <s xml:id="echoid-s14927" xml:space="preserve">φξ = R - {AD x DO/LO}.</s>
  <s xml:id="echoid-s14928" xml:space="preserve"/>
</p>
<pb o="124" file="0302" n="317" rhead=""/>
</div>
<div xml:id="echoid-div506" type="section" level="1" n="61">
<head xml:id="echoid-head64" xml:space="preserve">_Probl_. IV.</head>
<p>
  <s xml:id="echoid-s14929" xml:space="preserve">Sit angulus BDHrectus, &amp; </s>
  <s xml:id="echoid-s14930" xml:space="preserve">BF ad DH parallela; </s>
  <s xml:id="echoid-s14931" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14932" xml:space="preserve">_aſymptotis_ <lb/>
<anchor type="note" xlink:label="note-0302-01a" xlink:href="note-0302-01"/>
DB, DH per F deſcripta ſit _hyperbola_ FXG; </s>
  <s xml:id="echoid-s14933" xml:space="preserve">item centro Ddeſcrip-<lb/>tus ſit circulus KZL; </s>
  <s xml:id="echoid-s14934" xml:space="preserve">ſit denuò<unsure/> curva AMB talis, ut in hac ſumpto <lb/>quocunque puncto M, &amp; </s>
  <s xml:id="echoid-s14935" xml:space="preserve">per hoc trajectâ rectâ DMZ, item ſumptâ <lb/>DI = DM; </s>
  <s xml:id="echoid-s14936" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14937" xml:space="preserve">ductâ IX ad BF parallelâ, ſit _ſpatium hyperbolicum_ <lb/>BFXI æquale duplo _circulari ſectori_ ZDK; </s>
  <s xml:id="echoid-s14938" xml:space="preserve">curvæ AMB tangens <lb/>ad M determinetur.</s>
  <s xml:id="echoid-s14939" xml:space="preserve"/>
</p>
<div xml:id="echoid-div506" type="float" level="2" n="1">
<note position="left" xlink:label="note-0302-01" xlink:href="note-0302-01a" xml:space="preserve">Fig. 187.</note>
</div>
<p>
  <s xml:id="echoid-s14940" xml:space="preserve">Ducatur DS ad DM perpendicularis; </s>
  <s xml:id="echoid-s14941" xml:space="preserve">ſitque DB x BF = Rq; <lb/></s>
  <s xml:id="echoid-s14942" xml:space="preserve">fiátque DK. </s>
  <s xml:id="echoid-s14943" xml:space="preserve">R:</s>
  <s xml:id="echoid-s14944" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s14945" xml:space="preserve">P; </s>
  <s xml:id="echoid-s14946" xml:space="preserve">tum DK. </s>
  <s xml:id="echoid-s14947" xml:space="preserve">P:</s>
  <s xml:id="echoid-s14948" xml:space="preserve">: DM. </s>
  <s xml:id="echoid-s14949" xml:space="preserve">DT; </s>
  <s xml:id="echoid-s14950" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s14951" xml:space="preserve">connecta-<lb/>tur TM; </s>
  <s xml:id="echoid-s14952" xml:space="preserve">hæc curvam AMB tanget.</s>
  <s xml:id="echoid-s14953" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14954" xml:space="preserve">Adnotetur curvæ AMB hanc eſſe proprietatatem; </s>
  <s xml:id="echoid-s14955" xml:space="preserve">ut DI ſit inter <lb/>DB, DO (vel DA) eodem ordine _media proportionalis Geometricè_, <lb/>quo arcus KZ inter _o_<unsure/> (ſeu nihilum) &amp; </s>
  <s xml:id="echoid-s14956" xml:space="preserve">arcum KL eſt medius _Arith-_ <lb/>_meticè_. </s>
  <s xml:id="echoid-s14957" xml:space="preserve">hoc eſt, ſi DI ſit numerus in ſerie _Geometricè proprtionalium_ <lb/>incipiente à DB, &amp; </s>
  <s xml:id="echoid-s14958" xml:space="preserve">terminatâ in DA; </s>
  <s xml:id="echoid-s14959" xml:space="preserve">ac _o_<unsure/>, KL ſint Logarithmi <lb/>ipſarum DB, DA; </s>
  <s xml:id="echoid-s14960" xml:space="preserve">erit KZLogarithmus ipſius DI. </s>
  <s xml:id="echoid-s14961" xml:space="preserve">Vel <lb/>retrò (prout vulgares _Logarithmi_ procedunt, ſi DI ſit numerus in <lb/>ſerie _Geometrica_ exorſa à DO, &amp; </s>
  <s xml:id="echoid-s14962" xml:space="preserve">deſinente in DB ac _o_<unsure/> ſit _Logarith-_ <lb/>_mus_ ipſius DO, &amp; </s>
  <s xml:id="echoid-s14963" xml:space="preserve">arcus LK ipſius DB, erit arcus LZ _Logarithmus_ <lb/>ipfius DI.</s>
  <s xml:id="echoid-s14964" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14965" xml:space="preserve">Quod ſi abſolutè conſtruatur curva AMB, ejúſque _tangens Me-_ <lb/>_chanicè_ deprehendatur, inde patet _hpperbolici ſpatii Cycliſmum_ dari, <lb/>vel _Circuli hyperboliſmum_.</s>
  <s xml:id="echoid-s14966" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s14967" xml:space="preserve">Hujuſce _Spiralis_ naturam, ac dimenſionem (ut &amp; </s>
  <s xml:id="echoid-s14968" xml:space="preserve">Spatii BDA di-<lb/>menſionem) luculentè proſecutus eſt præclariſſimus D. </s>
  <s xml:id="echoid-s14969" xml:space="preserve">_Walliſſius<unsure/>_, in <lb/>Libro dè<unsure/> _Cycloide_; </s>
  <s xml:id="echoid-s14970" xml:space="preserve">quapropter de illa plura reticeo.</s>
  <s xml:id="echoid-s14971" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div508" type="section" level="1" n="62">
<head xml:id="echoid-head65" xml:space="preserve">_Probl_. V.</head>
<p>
  <s xml:id="echoid-s14972" xml:space="preserve">Sit ſpatium quodpiam EDG (rectis DE, DG, &amp; </s>
  <s xml:id="echoid-s14973" xml:space="preserve">linea ENG <lb/>
<anchor type="note" xlink:label="note-0302-02a" xlink:href="note-0302-02"/>
comprehenſa) &amp; </s>
  <s xml:id="echoid-s14974" xml:space="preserve">data quædam R; </s>
  <s xml:id="echoid-s14975" xml:space="preserve">curva AMB reperiatur talis, u <lb/>ſi utcunque à D projiciatur recta DNM, &amp; </s>
  <s xml:id="echoid-s14976" xml:space="preserve">DT ad hanc perpendi<emph style="sub">t</emph><unsure/> <lb/>cularis ſit, &amp; </s>
  <s xml:id="echoid-s14977" xml:space="preserve">MT curvam AMB contingat; </s>
  <s xml:id="echoid-s14978" xml:space="preserve">ſit DT. </s>
  <s xml:id="echoid-s14979" xml:space="preserve">DM:</s>
  <s xml:id="echoid-s14980" xml:space="preserve">: R-<lb/>DN.</s>
  <s xml:id="echoid-s14981" xml:space="preserve"/>
</p>
<div xml:id="echoid-div508" type="float" level="2" n="1">
<note position="left" xlink:label="note-0302-02" xlink:href="note-0302-02a" xml:space="preserve">Fig. 188.</note>
</div>
<p>
  <s xml:id="echoid-s14982" xml:space="preserve">Sit curva KZL talis, ut DZ = √ R x DN; </s>
  <s xml:id="echoid-s14983" xml:space="preserve">ſumptâque liberè
<pb o="125" file="0303" n="318" rhead=""/>
rectâ DB, ſit DB. </s>
  <s xml:id="echoid-s14984" xml:space="preserve">R:</s>
  <s xml:id="echoid-s14985" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s14986" xml:space="preserve">BF (ſit autem BF, ut &amp; </s>
  <s xml:id="echoid-s14987" xml:space="preserve">DHipſi DB <lb/>perpendicularis) tum per F, angulo BDHincluſa, tranſeat _hyperbola_ <lb/>FXX; </s>
  <s xml:id="echoid-s14988" xml:space="preserve">ſitque ſpatium BFXI (poſitâ nempe IX ad B<unsure/>F _parallelâ_) <lb/>æquale duplo ſpatio ZDL; </s>
  <s xml:id="echoid-s14989" xml:space="preserve">ſit denuò DM = DG; </s>
  <s xml:id="echoid-s14990" xml:space="preserve">erit Min cur-<lb/>va quæſita; </s>
  <s xml:id="echoid-s14991" xml:space="preserve">quam utique ſi tangat recta TM, erit TD. </s>
  <s xml:id="echoid-s14992" xml:space="preserve">DM:</s>
  <s xml:id="echoid-s14993" xml:space="preserve">: R. <lb/></s>
  <s xml:id="echoid-s14994" xml:space="preserve">DN.</s>
  <s xml:id="echoid-s14995" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div510" type="section" level="1" n="63">
<head xml:id="echoid-head66" xml:space="preserve">_Probl_. VI.</head>
<p>
  <s xml:id="echoid-s14996" xml:space="preserve">Sit rurſus ſpatium EDG (ut in præcedente) reperienda eſt curva <lb/>AMB, ad quam ſi projiciatur recta DNM, &amp; </s>
  <s xml:id="echoid-s14997" xml:space="preserve">ſit DT huic perpen-<lb/>
<anchor type="note" xlink:label="note-0303-01a" xlink:href="note-0303-01"/>
dicularis, &amp; </s>
  <s xml:id="echoid-s14998" xml:space="preserve">MT curvam AMB tangat, fuerit DT = DN.</s>
  <s xml:id="echoid-s14999" xml:space="preserve"/>
</p>
<div xml:id="echoid-div510" type="float" level="2" n="1">
<note position="right" xlink:label="note-0303-01" xlink:href="note-0303-01a" xml:space="preserve">Fig. 188.</note>
</div>
<p>
  <s xml:id="echoid-s15000" xml:space="preserve">Adſumatur quæpiam R, &amp; </s>
  <s xml:id="echoid-s15001" xml:space="preserve">ſit DZ q = {R<emph style="sub">3</emph>/DN}; </s>
  <s xml:id="echoid-s15002" xml:space="preserve">item acceptâ DB <lb/>(cui perpendiculares DH, BF = {R<emph style="sub">3</emph>/DBq}; </s>
  <s xml:id="echoid-s15003" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15004" xml:space="preserve">per F intra _aſymptotos_ <lb/>DB, DH deſcribatur _hyperboliformis_ ſecundi generis (in qua nempe <lb/>ordinatæ, ceu BF, vel IX, ſint quartæ proportionales in ratione DB <lb/>ad R, vel DG ad R) tum capiatur ſpatium BIXF æquale duplo <lb/>ZDL; </s>
  <s xml:id="echoid-s15005" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15006" xml:space="preserve">ſit DM = DI; </s>
  <s xml:id="echoid-s15007" xml:space="preserve">erit M in curva quæſita; </s>
  <s xml:id="echoid-s15008" xml:space="preserve">quam ſi tan-<lb/>gat MT, erit DT = DN.</s>
  <s xml:id="echoid-s15009" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div512" type="section" level="1" n="64">
<head xml:id="echoid-head67" xml:space="preserve">_Probl_. VII</head>
<p>
  <s xml:id="echoid-s15010" xml:space="preserve">Sit figura quævis ADB (cujus _axis_ AD, _baſis_ DB) &amp; </s>
  <s xml:id="echoid-s15011" xml:space="preserve">utcunque <lb/>
<anchor type="note" xlink:label="note-0303-02a" xlink:href="note-0303-02"/>
ductâ PM ad DB parallelâ datum ſit (ſeu expreſſum quomodocunque) <lb/>ſpatium APM, oportet hinc ordinatam PM exhibere, vel expri-<lb/>mere.</s>
  <s xml:id="echoid-s15012" xml:space="preserve"/>
</p>
<div xml:id="echoid-div512" type="float" level="2" n="1">
<note position="right" xlink:label="note-0303-02" xlink:href="note-0303-02a" xml:space="preserve">Fig. 189.</note>
</div>
<p>
  <s xml:id="echoid-s15013" xml:space="preserve">Acceptâ quâqiam R, ſit R x PZ = APM; </s>
  <s xml:id="echoid-s15014" xml:space="preserve">hinc emergat linea <lb/>AZZK; </s>
  <s xml:id="echoid-s15015" xml:space="preserve">huic perpendicularis reperiatur ZO; </s>
  <s xml:id="echoid-s15016" xml:space="preserve">tum erit PZPO <lb/>:</s>
  <s xml:id="echoid-s15017" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s15018" xml:space="preserve">PM.</s>
  <s xml:id="echoid-s15019" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15020" xml:space="preserve">_Exemp_. </s>
  <s xml:id="echoid-s15021" xml:space="preserve">AP vocetur x &amp; </s>
  <s xml:id="echoid-s15022" xml:space="preserve">ſit APM = √ r x<emph style="sub">3</emph>, ergo PZ = √ <lb/>{x<emph style="sub">3</emph>/r}; </s>
  <s xml:id="echoid-s15023" xml:space="preserve">unde reperietur PO = {3 x x/2 r}. </s>
  <s xml:id="echoid-s15024" xml:space="preserve">Eſtque √ {x<emph style="sub">3</emph>/r}. </s>
  <s xml:id="echoid-s15025" xml:space="preserve">{3 x x/2 r} <lb/>:</s>
  <s xml:id="echoid-s15026" xml:space="preserve">: r. </s>
  <s xml:id="echoid-s15027" xml:space="preserve">{3/2} √ r x = PM. </s>
  <s xml:id="echoid-s15028" xml:space="preserve">unde AMB eſt _Parabola_, cujus _Pa-_ <lb/>rameter eſt {9/4} r.</s>
  <s xml:id="echoid-s15029" xml:space="preserve"/>
</p>
<pb o="126" file="0304" n="319" rhead=""/>
<p>
  <s xml:id="echoid-s15030" xml:space="preserve">_Aliter_. </s>
  <s xml:id="echoid-s15031" xml:space="preserve">Fiat PZ = √ 2 APM. </s>
  <s xml:id="echoid-s15032" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15033" xml:space="preserve">ſit ZO curvæ AZK perpendi-<lb/>cularis; </s>
  <s xml:id="echoid-s15034" xml:space="preserve">erit PM = PO.</s>
  <s xml:id="echoid-s15035" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15036" xml:space="preserve">_Exemp_. </s>
  <s xml:id="echoid-s15037" xml:space="preserve">Sit AP = x; </s>
  <s xml:id="echoid-s15038" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15039" xml:space="preserve">APM = {x<emph style="sub">3</emph>/r}. </s>
  <s xml:id="echoid-s15040" xml:space="preserve">quare PZ = √ {2 x<emph style="sub">3</emph>/r} <lb/>unde reperietur PO = {3 x x/r} = PM; </s>
  <s xml:id="echoid-s15041" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15042" xml:space="preserve">rurſus AMB <lb/>erit _Parabola_.</s>
  <s xml:id="echoid-s15043" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div514" type="section" level="1" n="65">
<head xml:id="echoid-head68" xml:space="preserve">_Probl_. VIII.</head>
<p>
  <s xml:id="echoid-s15044" xml:space="preserve">Sit figura quævis ADB (rectis DA, DB, &amp; </s>
  <s xml:id="echoid-s15045" xml:space="preserve">linea AMB com-<lb/>
<anchor type="note" xlink:label="note-0304-01a" xlink:href="note-0304-01"/>
prehenſa) &amp; </s>
  <s xml:id="echoid-s15046" xml:space="preserve">à Dutcunque projectâ rectâ DM, datum ſit ſpatium <lb/>ADM; </s>
  <s xml:id="echoid-s15047" xml:space="preserve">oportet rectam DM definire.</s>
  <s xml:id="echoid-s15048" xml:space="preserve"/>
</p>
<div xml:id="echoid-div514" type="float" level="2" n="1">
<note position="left" xlink:label="note-0304-01" xlink:href="note-0304-01a" xml:space="preserve">Fig. 190.</note>
</div>
<p>
  <s xml:id="echoid-s15049" xml:space="preserve">Acceptâ quâpiam R, ſit DZ = {2 ADM/R}; </s>
  <s xml:id="echoid-s15050" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15051" xml:space="preserve">ZO curvæ AZK <lb/>perpendicularis; </s>
  <s xml:id="echoid-s15052" xml:space="preserve">cui occurrat DH ad DM perpendicularis; <lb/></s>
  <s xml:id="echoid-s15053" xml:space="preserve">erit DM = √ R x DO.</s>
  <s xml:id="echoid-s15054" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15055" xml:space="preserve">_Aliter_. </s>
  <s xml:id="echoid-s15056" xml:space="preserve">Sit DZ = √ 4 ADM; </s>
  <s xml:id="echoid-s15057" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15058" xml:space="preserve">ZO curvæ AZK perpen-<lb/>dicularis; </s>
  <s xml:id="echoid-s15059" xml:space="preserve">cui occurrat DH ad DZ perpendicularis; </s>
  <s xml:id="echoid-s15060" xml:space="preserve">erit DM <lb/> = √ DZ x DO.</s>
  <s xml:id="echoid-s15061" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15062" xml:space="preserve">_De figuris involutis &amp; </s>
  <s xml:id="echoid-s15063" xml:space="preserve">evolutis_ bellam σκέψιγ inſtituit _Præclarus Ge-_ <lb/>_ometra D. </s>
  <s xml:id="echoid-s15064" xml:space="preserve">Gregorius Aberd._ </s>
  <s xml:id="echoid-s15065" xml:space="preserve">Alienæ meſſi nollem ego falcem meam <lb/>immittere, verùm liceat utcunque iſthuc pertinentes (aliud agenti quæ <lb/>mihi ſe ingeſſerunt) unam aut alteram obſervatiunculam his intexere.</s>
  <s xml:id="echoid-s15066" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div516" type="section" level="1" n="66">
<head xml:id="echoid-head69" xml:space="preserve">_Probl_. IX.</head>
<p>
  <s xml:id="echoid-s15067" xml:space="preserve">Data fit figura quæpiam ADB (cujus _axis_ AD, _baſis_ DB) oper-<lb/>
<anchor type="note" xlink:label="note-0304-02a" xlink:href="note-0304-02"/>
tet ei congruentem involutam exhibere.</s>
  <s xml:id="echoid-s15068" xml:space="preserve"/>
</p>
<div xml:id="echoid-div516" type="float" level="2" n="1">
<note position="left" xlink:label="note-0304-02" xlink:href="note-0304-02a" xml:space="preserve">Fig. 191.</note>
</div>
<p>
  <s xml:id="echoid-s15069" xml:space="preserve">_Centro_ C, intervallo quopiam CL deſcribatur _Circulus_ LXX; </s>
  <s xml:id="echoid-s15070" xml:space="preserve">ſit <lb/>
<anchor type="note" xlink:label="note-0304-03a" xlink:href="note-0304-03"/>
autem curva KZZ talis, ut pro lubitu ductâ rectâ MPZ ad BD pa-
<pb o="127" file="0305" n="320" rhead=""/>
rallelâ, ſit rectangulum ex PM, PZ æquale quadrato ex CL (vel <lb/>PZ = {CL q/PM}). </s>
  <s xml:id="echoid-s15071" xml:space="preserve">Sit tum arc. </s>
  <s xml:id="echoid-s15072" xml:space="preserve">LX = {ſpat. </s>
  <s xml:id="echoid-s15073" xml:space="preserve">DKZP/CL} (vel ſector <lb/>LCX ſubduplw<unsure/>s ſpatii DKZP) &amp; </s>
  <s xml:id="echoid-s15074" xml:space="preserve">in CX capiatur C μ = PM; <lb/></s>
  <s xml:id="echoid-s15075" xml:space="preserve">erit linea βμμ ipſius BMA involuta; </s>
  <s xml:id="echoid-s15076" xml:space="preserve">vel ſpatium Cμβ ſpatii <lb/>ADB.)</s>
  <s xml:id="echoid-s15077" xml:space="preserve"/>
</p>
<div xml:id="echoid-div517" type="float" level="2" n="2">
<note position="left" xlink:label="note-0304-03" xlink:href="note-0304-03a" xml:space="preserve">Fig. 192.</note>
</div>
<p>
  <s xml:id="echoid-s15078" xml:space="preserve">_Exemp_. </s>
  <s xml:id="echoid-s15079" xml:space="preserve">Sit ADB circuli quadrans; </s>
  <s xml:id="echoid-s15080" xml:space="preserve">erit ergò (quod è præmonſtra-<lb/>tis conſtat) ſpat. </s>
  <s xml:id="echoid-s15081" xml:space="preserve">DKZP (2 ſector LCX). </s>
  <s xml:id="echoid-s15082" xml:space="preserve">ſect. </s>
  <s xml:id="echoid-s15083" xml:space="preserve">BDM <lb/>:</s>
  <s xml:id="echoid-s15084" xml:space="preserve">: CLq. </s>
  <s xml:id="echoid-s15085" xml:space="preserve">DBq. </s>
  <s xml:id="echoid-s15086" xml:space="preserve">unde arc. </s>
  <s xml:id="echoid-s15087" xml:space="preserve">LX. </s>
  <s xml:id="echoid-s15088" xml:space="preserve">arc. </s>
  <s xml:id="echoid-s15089" xml:space="preserve">BM:</s>
  <s xml:id="echoid-s15090" xml:space="preserve">: CL. </s>
  <s xml:id="echoid-s15091" xml:space="preserve">DB. <lb/></s>
  <s xml:id="echoid-s15092" xml:space="preserve">quare ang. </s>
  <s xml:id="echoid-s15093" xml:space="preserve">LCX = ang. </s>
  <s xml:id="echoid-s15094" xml:space="preserve">BDM = ang. </s>
  <s xml:id="echoid-s15095" xml:space="preserve">DMP. </s>
  <s xml:id="echoid-s15096" xml:space="preserve">unde ang. </s>
  <s xml:id="echoid-s15097" xml:space="preserve"><lb/>C μβ eſt rectus, adeóque linea βμ C eſt _ſemicirculus_.</s>
  <s xml:id="echoid-s15098" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15099" xml:space="preserve">_Coroll_. </s>
  <s xml:id="echoid-s15100" xml:space="preserve">1. </s>
  <s xml:id="echoid-s15101" xml:space="preserve">Subnotari poteſt, ſi duæ ſiguræ ADB, ADG analogæ fu-<lb/>
<anchor type="note" xlink:label="note-0305-01a" xlink:href="note-0305-01"/>
erint; </s>
  <s xml:id="echoid-s15102" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15103" xml:space="preserve">harum _involu<unsure/>tæ_ ſint _Cμβ Cνγ_; </s>
  <s xml:id="echoid-s15104" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15105" xml:space="preserve">fuerit _Cμ. </s>
  <s xml:id="echoid-s15106" xml:space="preserve">Cν_ <lb/>:</s>
  <s xml:id="echoid-s15107" xml:space="preserve">: DB. </s>
  <s xml:id="echoid-s15108" xml:space="preserve">DG; </s>
  <s xml:id="echoid-s15109" xml:space="preserve">erit reciprocè ang. </s>
  <s xml:id="echoid-s15110" xml:space="preserve">_βCμ. </s>
  <s xml:id="echoid-s15111" xml:space="preserve">β Cν:</s>
  <s xml:id="echoid-s15112" xml:space="preserve">: DG_. <lb/></s>
  <s xml:id="echoid-s15113" xml:space="preserve">DB.</s>
  <s xml:id="echoid-s15114" xml:space="preserve"/>
</p>
<div xml:id="echoid-div518" type="float" level="2" n="3">
<note position="right" xlink:label="note-0305-01" xlink:href="note-0305-01a" xml:space="preserve">Fig. 193.</note>
</div>
<p>
  <s xml:id="echoid-s15115" xml:space="preserve">2. </s>
  <s xml:id="echoid-s15116" xml:space="preserve">Illud etiam conversè valet.</s>
  <s xml:id="echoid-s15117" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15118" xml:space="preserve">3. </s>
  <s xml:id="echoid-s15119" xml:space="preserve">Sin curvæ Cνγ, CS β ſuo modo analogæ fuerint, hoc eſt, <lb/>
<anchor type="note" xlink:label="note-0305-02a" xlink:href="note-0305-02"/>
ſi utcunque à Cprojectâ rectâ C ν S, habeant Cν, CS ean-<lb/>dem perpetuò rationem, erunt hæ ſimilium linearum _invo-_ <lb/>_lutæ_.</s>
  <s xml:id="echoid-s15120" xml:space="preserve"/>
</p>
<div xml:id="echoid-div519" type="float" level="2" n="4">
<note position="right" xlink:label="note-0305-02" xlink:href="note-0305-02a" xml:space="preserve">Fig. 194.</note>
</div>
</div>
<div xml:id="echoid-div521" type="section" level="1" n="67">
<head xml:id="echoid-head70" xml:space="preserve">_Probl_. X.</head>
<p>
  <s xml:id="echoid-s15121" xml:space="preserve">Dàta figurâ quâpiam β C φ rectis C β, C φ, &amp; </s>
  <s xml:id="echoid-s15122" xml:space="preserve">aliâ lineâ βφ <lb/>
<anchor type="note" xlink:label="note-0305-03a" xlink:href="note-0305-03"/>
comprehensâ, eicompetentem _evolutam_ deſignare.</s>
  <s xml:id="echoid-s15123" xml:space="preserve"/>
</p>
<div xml:id="echoid-div521" type="float" level="2" n="1">
<note position="right" xlink:label="note-0305-03" xlink:href="note-0305-03a" xml:space="preserve">Fig. 195.</note>
</div>
<p>
  <s xml:id="echoid-s15124" xml:space="preserve">_Centro_ Cutcunque deſcribatur _circularis arcus_ LE (cum rectis Cβ, <lb/>Cφ conſtituens ſectorem LCE) tum ductâ CK ad LC perpendicu-<lb/>
<anchor type="note" xlink:label="note-0305-04a" xlink:href="note-0305-04"/>
lari, ſit curva β YH ità rectam CK reſpiciens, ut liberè projectâ rectà <lb/>CμZ, ſumptâque CO = arcLZ, ductâque OY ad CK perpen-<lb/>diculari, ſitOY = Cμ; </s>
  <s xml:id="echoid-s15125" xml:space="preserve">porrò ad rectam DA ſic referatur curva <lb/>BMF, ut cùm ſit DP = {ſpat. </s>
  <s xml:id="echoid-s15126" xml:space="preserve">C β YO/CL}; </s>
  <s xml:id="echoid-s15127" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15128" xml:space="preserve">PM ad DA perpendi-<lb/>cularis; </s>
  <s xml:id="echoid-s15129" xml:space="preserve">ſit eti<unsure/>am PM = Cμ; </s>
  <s xml:id="echoid-s15130" xml:space="preserve">erit ſpatium DBFA ipſins Cβφ _evolutum_.</s>
  <s xml:id="echoid-s15131" xml:space="preserve"/>
</p>
<div xml:id="echoid-div522" type="float" level="2" n="2">
<note position="right" xlink:label="note-0305-04" xlink:href="note-0305-04a" xml:space="preserve">Fig. 196.</note>
</div>
<p>
  <s xml:id="echoid-s15132" xml:space="preserve">_Exemp_. </s>
  <s xml:id="echoid-s15133" xml:space="preserve">Sit LZE arcus circuli centro C deſcripti, &amp; </s>
  <s xml:id="echoid-s15134" xml:space="preserve">βμ C ejuſmodi <lb/>
<anchor type="note" xlink:label="note-0305-05a" xlink:href="note-0305-05"/>
<pb o="128" file="0306" n="321" rhead=""/>
_ſpiralis_, ut pro arbitrio ductâ rectâ C μ Z habeat arcus EZ ad rectam <lb/>C μ rationem aſſignatam (puta R ad S) Manifeſtum eſt lineam β YH <lb/>eſſe rectam, quoniam EZ (KO). </s>
  <s xml:id="echoid-s15135" xml:space="preserve">Cμ (OY):</s>
  <s xml:id="echoid-s15136" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s15137" xml:space="preserve">S, perpetuò. <lb/></s>
  <s xml:id="echoid-s15138" xml:space="preserve">
<anchor type="note" xlink:label="note-0306-01a" xlink:href="note-0306-01"/>
unde evoluta BMF ſit _Parabola_; </s>
  <s xml:id="echoid-s15139" xml:space="preserve">quoniam axis partes AP, AD ſe <lb/>habent ut ſpatia KOY, KC β, hoc eſt ut quadrata ex ipſis OY, Cβ, <lb/>vel ex ipſis PM, DB.</s>
  <s xml:id="echoid-s15140" xml:space="preserve"/>
</p>
<div xml:id="echoid-div523" type="float" level="2" n="3">
<note position="right" xlink:label="note-0305-05" xlink:href="note-0305-05a" xml:space="preserve">Fig. 197.</note>
<note position="left" xlink:label="note-0306-01" xlink:href="note-0306-01a" xml:space="preserve">Fig. 198.</note>
</div>
</div>
<div xml:id="echoid-div525" type="section" level="1" n="68">
<head xml:id="echoid-head71" xml:space="preserve">_Corol. Theor_. I.</head>
<p>
  <s xml:id="echoid-s15141" xml:space="preserve">Si ad figuram βCφ erigatur _cylindricus_ altitudinem habens æqua-<lb/>lem peripheriæ integræ _circuli_, cujus radius CL; </s>
  <s xml:id="echoid-s15142" xml:space="preserve">erit iſte _cylindricus_ <lb/>æ<unsure/>qualis _ſolido_, quod procreatur è figurâ Cβ HK circa axem CK ro-<lb/>tatâ.</s>
  <s xml:id="echoid-s15143" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div526" type="section" level="1" n="69">
<head xml:id="echoid-head72" xml:space="preserve">_Theor_. II.</head>
<p>
  <s xml:id="echoid-s15144" xml:space="preserve">Sit curva quæpiam AMB (cujus axis AD, baſis DB) &amp; </s>
  <s xml:id="echoid-s15145" xml:space="preserve">curva <lb/>
<anchor type="note" xlink:label="note-0306-02a" xlink:href="note-0306-02"/>
AZL talis, ut liberè ductâ rectâ ZPM, ſit PZ = √ 2 APM; </s>
  <s xml:id="echoid-s15146" xml:space="preserve">ſit <lb/>item alia curva OYY talis, ut ad hanc productâ rectâ ZPMY, <lb/>adſumptâque rectâ R, ſit ZP q. </s>
  <s xml:id="echoid-s15147" xml:space="preserve">R q:</s>
  <s xml:id="echoid-s15148" xml:space="preserve">: PM. </s>
  <s xml:id="echoid-s15149" xml:space="preserve">PY; </s>
  <s xml:id="echoid-s15150" xml:space="preserve">ſitque denuò DL. <lb/></s>
  <s xml:id="echoid-s15151" xml:space="preserve">R:</s>
  <s xml:id="echoid-s15152" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s15153" xml:space="preserve">LE. </s>
  <s xml:id="echoid-s15154" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15155" xml:space="preserve">per E intra angulum LDG deſcribatur _Hyper-_ <lb/>
<anchor type="note" xlink:label="note-0306-03a" xlink:href="note-0306-03"/>
_bola_ EXX; </s>
  <s xml:id="echoid-s15156" xml:space="preserve">huic autem occurrat ducta recta ZHX ad AD parallela, <lb/>erit ſpatium PDOY æquale _ſpatio Hyperbolico_ LHXE.</s>
  <s xml:id="echoid-s15157" xml:space="preserve"/>
</p>
<div xml:id="echoid-div526" type="float" level="2" n="1">
<note position="left" xlink:label="note-0306-02" xlink:href="note-0306-02a" xml:space="preserve">Fig. 195.</note>
<note position="left" xlink:label="note-0306-03" xlink:href="note-0306-03a" xml:space="preserve">Fig. 199.</note>
</div>
<p>
  <s xml:id="echoid-s15158" xml:space="preserve">Hinc _ſumma_ omnium {PM/APM} = {2 LEXH/R q}.</s>
  <s xml:id="echoid-s15159" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div528" type="section" level="1" n="70">
<head xml:id="echoid-head73" xml:space="preserve">_Theor_. III.</head>
<p>
  <s xml:id="echoid-s15160" xml:space="preserve">Sit curva quæpiam AMB, cujus axis AD, baſis DB; </s>
  <s xml:id="echoid-s15161" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15162" xml:space="preserve">curva <lb/>KZL talis, ut adſumptâ quâdam R, &amp; </s>
  <s xml:id="echoid-s15163" xml:space="preserve">arbitrariè ductâ rectâ ZPM <lb/>ad BD parallelâ, ſit √ APM. </s>
  <s xml:id="echoid-s15164" xml:space="preserve">PM:</s>
  <s xml:id="echoid-s15165" xml:space="preserve">: R. </s>
  <s xml:id="echoid-s15166" xml:space="preserve">PZ; </s>
  <s xml:id="echoid-s15167" xml:space="preserve">erit ſpatium ADLK <lb/>
<anchor type="note" xlink:label="note-0306-04a" xlink:href="note-0306-04"/>
æquale _rectangulo_ ex R in 2 √ ADB; </s>
  <s xml:id="echoid-s15168" xml:space="preserve">vel {ADLK/2 R} = √ ADB.</s>
  <s xml:id="echoid-s15169" xml:space="preserve"/>
</p>
<div xml:id="echoid-div528" type="float" level="2" n="1">
<note position="left" xlink:label="note-0306-04" xlink:href="note-0306-04a" xml:space="preserve">Fig. 200.</note>
</div>
<p>
  <s xml:id="echoid-s15170" xml:space="preserve">_Exemp_. </s>
  <s xml:id="echoid-s15171" xml:space="preserve">Sit ADB circuli quadrans, erit ſumma omnium {PM/APM} = <lb/>√ 2 DA x arc. </s>
  <s xml:id="echoid-s15172" xml:space="preserve">AB.</s>
  <s xml:id="echoid-s15173" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div530" type="section" level="1" n="71">
<head xml:id="echoid-head74" xml:space="preserve">_Theor_. IV.</head>
<p>
  <s xml:id="echoid-s15174" xml:space="preserve">Sit curva quæpiam AMB (cujus axis AD, baſis DB) ſintque <lb/>duæ lineæ EXK, GYL ità relatæ, ut in curva AMB ſumpto quopi-
<pb o="129" file="0307" n="322" rhead=""/>
am puncto M, ductíſque rectis MPX ad BD, &amp; </s>
  <s xml:id="echoid-s15175" xml:space="preserve">MQY ad AD <lb/>parallelis, poſitóque rectam MT tangere curvam AMB, ſit TP. <lb/></s>
  <s xml:id="echoid-s15176" xml:space="preserve">
<anchor type="note" xlink:label="note-0307-01a" xlink:href="note-0307-01"/>
PM:</s>
  <s xml:id="echoid-s15177" xml:space="preserve">: QY. </s>
  <s xml:id="echoid-s15178" xml:space="preserve">PX; </s>
  <s xml:id="echoid-s15179" xml:space="preserve">erunt figuræ ADKE, DBLG ſibimet æqua-<lb/>les.</s>
  <s xml:id="echoid-s15180" xml:space="preserve"/>
</p>
<div xml:id="echoid-div530" type="float" level="2" n="1">
<note position="right" xlink:label="note-0307-01" xlink:href="note-0307-01a" xml:space="preserve">Fig.201.</note>
</div>
<p>
  <s xml:id="echoid-s15181" xml:space="preserve">Valet hoc converſum. </s>
  <s xml:id="echoid-s15182" xml:space="preserve">Nempe ſi figuræ ADKE, DBLG æ-<lb/>quentur, &amp; </s>
  <s xml:id="echoid-s15183" xml:space="preserve">MT curvam AMB tangat, erit TP. </s>
  <s xml:id="echoid-s15184" xml:space="preserve">PM:</s>
  <s xml:id="echoid-s15185" xml:space="preserve">: QY. <lb/></s>
  <s xml:id="echoid-s15186" xml:space="preserve">PX:</s>
  <s xml:id="echoid-s15187" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15188" xml:space="preserve">_Not_. </s>
  <s xml:id="echoid-s15189" xml:space="preserve">Omnium hactenus Propoſitorum fœcundiſſimum eſt hoc <lb/>_Tbeorema_; </s>
  <s xml:id="echoid-s15190" xml:space="preserve">præcedentium quippe complura vel in eo continentur, aut <lb/>ab eo facilè conſectantur. </s>
  <s xml:id="echoid-s15191" xml:space="preserve">Nam poſito lineam AMB indeterminatam <lb/>eſſe naturâ, ſi ipſarum EXK, GYL alterutra pro tuo arbitratu de-<lb/>terminetur, exinde reſultabit Theorema quoddam ejuſmodi, qualia <lb/>ſuperiùs exhibentur aliquammulta. </s>
  <s xml:id="echoid-s15192" xml:space="preserve">Si _e. </s>
  <s xml:id="echoid-s15193" xml:space="preserve">g_. </s>
  <s xml:id="echoid-s15194" xml:space="preserve">linea GYL ponatur recta <lb/>cum ipſa BD ſemi-rectum conſtituens angulum (quo caſu concipiun-<lb/>tur puncta D, G coincidere) proveniet indè prima _Lectionis_ XI. </s>
  <s xml:id="echoid-s15195" xml:space="preserve">Si <lb/>GYL ſit recta ad DB parallela, emerget _Lectionis ejuſdem._ </s>
  <s xml:id="echoid-s15196" xml:space="preserve">Rur-<lb/>
<anchor type="note" xlink:label="note-0307-02a" xlink:href="note-0307-02"/>
ſus ſi PM = PX (vel lineæ AMB, EXK ſint eædem) conſeque-<lb/>tur hinc _decima_ ejuſdem. </s>
  <s xml:id="echoid-s15197" xml:space="preserve">Exhinc porrò liquet adſumpto cuilibet ſpa-<lb/>tio _infinita, genere diverſa, ſpatia æqualia_ facilè deſignari veluti ſi _ſpa-_ <lb/>_tium_ DGLB ponatur _circuli quadrans_, cujus _centrum_ D; </s>
  <s xml:id="echoid-s15198" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15199" xml:space="preserve">curva <lb/>AMB ſit _parabola_, cujus _axis_ AD, emerget curvæ EXK hæc pro-<lb/>prietas, ut (ſi dicatur DB = r; </s>
  <s xml:id="echoid-s15200" xml:space="preserve">AP = x; </s>
  <s xml:id="echoid-s15201" xml:space="preserve">PX = y; </s>
  <s xml:id="echoid-s15202" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15203" xml:space="preserve">_k_ (vel <lb/>{DB q/2 AD}) ſit _parabolæ ſemipar ameter_) ſit {_rrk_/2} = _kkx_ + _xyy_. </s>
  <s xml:id="echoid-s15204" xml:space="preserve">Sin <lb/>AMB ponatur _hyperbola_, procreabitur alterius generis curva EXK. <lb/></s>
  <s xml:id="echoid-s15205" xml:space="preserve">his autem expenſis ἀβλεφι<unsure/>αν meam incuſo, qui non hoc _Theorema_ (ſi-<lb/>cut &amp; </s>
  <s xml:id="echoid-s15206" xml:space="preserve">ea quæ ſubſequuntur, quorum ferè ratio conſimilis eſt, &amp; </s>
  <s xml:id="echoid-s15207" xml:space="preserve">ſup-<lb/>par uſus) primo loco poſuerim, &amp; </s>
  <s xml:id="echoid-s15208" xml:space="preserve">ex eo (nec non è reliquis mox <lb/>ſubjiciendis) quod fieri poſſe video, reliqua deduxerim. </s>
  <s xml:id="echoid-s15209" xml:space="preserve">Veruntamen <lb/>hujuſmodi _Phrygiam ſapientiam_ juxta mecum pleriſque familiarem au-<lb/>tumo, literas has tractantibus.</s>
  <s xml:id="echoid-s15210" xml:space="preserve"/>
</p>
<div xml:id="echoid-div531" type="float" level="2" n="2">
<note position="right" xlink:label="note-0307-02" xlink:href="note-0307-02a" xml:space="preserve">Fig. 202.</note>
</div>
</div>
<div xml:id="echoid-div533" type="section" level="1" n="72">
<head xml:id="echoid-head75" xml:space="preserve">_Theor_. V.</head>
<p>
  <s xml:id="echoid-s15211" xml:space="preserve">Sit ſpatium quodpiam ADB (rectis DA, DB, &amp; </s>
  <s xml:id="echoid-s15212" xml:space="preserve">curva AMB <lb/>
<anchor type="note" xlink:label="note-0307-03a" xlink:href="note-0307-03"/>
comprehenſum) ſint item curvæ EXK, GYL ità relatæ, ut ſi in curva <lb/>AMB liberè ſumatur punctum M, ducatur DMX, ſit DQ = DM, <lb/>ducatur QY ad DB perdendicularis, ſit DT ad DM perpendicula-<lb/>ris, recta MT curvam AMB contingat; </s>
  <s xml:id="echoid-s15213" xml:space="preserve">ſi, his inquam ſuppoſitis, ſit <lb/>TD. </s>
  <s xml:id="echoid-s15214" xml:space="preserve">DM:</s>
  <s xml:id="echoid-s15215" xml:space="preserve">: DM x QY. </s>
  <s xml:id="echoid-s15216" xml:space="preserve">DXq; </s>
  <s xml:id="echoid-s15217" xml:space="preserve">erit ſpatium DGLB ſpatii EDK <lb/>duplum.</s>
  <s xml:id="echoid-s15218" xml:space="preserve"/>
</p>
<div xml:id="echoid-div533" type="float" level="2" n="1">
<note position="right" xlink:label="note-0307-03" xlink:href="note-0307-03a" xml:space="preserve">Fig. 203.</note>
</div>
<pb o="130" file="0308" n="323" rhead=""/>
</div>
<div xml:id="echoid-div535" type="section" level="1" n="73">
<head xml:id="echoid-head76" xml:space="preserve">_Theor_. VI.</head>
<p>
  <s xml:id="echoid-s15219" xml:space="preserve">Sit rurſus AMB curva quævis (cujus axis AD, baſis DB) &amp; </s>
  <s xml:id="echoid-s15220" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0308-01a" xlink:href="note-0308-01"/>
curvæ EXK, HZO ita verſus ſe, &amp; </s>
  <s xml:id="echoid-s15221" xml:space="preserve">axes AD, αβ relatæ, ut arbi-<lb/>trariè in curva AMB accepto puncto M, &amp; </s>
  <s xml:id="echoid-s15222" xml:space="preserve">ductâ MPX ad AD per-<lb/>pendiculari, ſumptâ αμ = arc AM, ductâ μZ ad αβ perpendiculari, <lb/>poſitóque rectam TM curvam AMB tangere; </s>
  <s xml:id="echoid-s15223" xml:space="preserve">ſit TP. </s>
  <s xml:id="echoid-s15224" xml:space="preserve">TM:</s>
  <s xml:id="echoid-s15225" xml:space="preserve">: μ Z. <lb/></s>
  <s xml:id="echoid-s15226" xml:space="preserve">PX; </s>
  <s xml:id="echoid-s15227" xml:space="preserve">erunt ſpatia ADKE, α β OH æqualia ſibi.</s>
  <s xml:id="echoid-s15228" xml:space="preserve"/>
</p>
<div xml:id="echoid-div535" type="float" level="2" n="1">
<note position="left" xlink:label="note-0308-01" xlink:href="note-0308-01a" xml:space="preserve">Fig. 204.</note>
</div>
</div>
<div xml:id="echoid-div537" type="section" level="1" n="74">
<head xml:id="echoid-head77" xml:space="preserve">_Theor_. VII.</head>
<p>
  <s xml:id="echoid-s15229" xml:space="preserve">Sit ſpatium quodpiam<unsure/> ADB (rectis DA, DB, &amp; </s>
  <s xml:id="echoid-s15230" xml:space="preserve">curvâ AMB <lb/>
<anchor type="note" xlink:label="note-0308-02a" xlink:href="note-0308-02"/>
definitum) ſint item curvæ EXK, HZO ità relatæ, ut ſi quodvis <lb/>capiatur punctum M in curva AMB, projiciatur recta DMX, ſuma-<lb/>tur αμ = arc AM; </s>
  <s xml:id="echoid-s15231" xml:space="preserve">ducatur μZ ad rectam αβ perpendicularis; </s>
  <s xml:id="echoid-s15232" xml:space="preserve">ſit <lb/>DT perpendicularis ipſi DM; </s>
  <s xml:id="echoid-s15233" xml:space="preserve">recta MT curvam AMB tangat; </s>
  <s xml:id="echoid-s15234" xml:space="preserve">ſit <lb/>TD. </s>
  <s xml:id="echoid-s15235" xml:space="preserve">TM:</s>
  <s xml:id="echoid-s15236" xml:space="preserve">: DM x μ Z. </s>
  <s xml:id="echoid-s15237" xml:space="preserve">DX q; </s>
  <s xml:id="echoid-s15238" xml:space="preserve">erit ſpatium αβ OH ſpatii EDK <lb/>duplum.</s>
  <s xml:id="echoid-s15239" xml:space="preserve"/>
</p>
<div xml:id="echoid-div537" type="float" level="2" n="1">
<note position="left" xlink:label="note-0308-02" xlink:href="note-0308-02a" xml:space="preserve">Fig. 204, <lb/>205.</note>
</div>
<p>
  <s xml:id="echoid-s15240" xml:space="preserve">Sed horum hic eſto terminus.</s>
  <s xml:id="echoid-s15241" xml:space="preserve"/>
</p>
<pb o="131" file="0309" n="324"/>
</div>
<div xml:id="echoid-div539" type="section" level="1" n="75">
<head xml:id="echoid-head78" xml:space="preserve"><emph style="sc">Lect</emph>. XIII.</head>
<p>
  <s xml:id="echoid-s15242" xml:space="preserve">Æ _Quationum_ naturam è terminorum _analogia_ expoſuit _Vieta_; <lb/></s>
  <s xml:id="echoid-s15243" xml:space="preserve">illam ex eorum in ſe ductu dilucidiùs explicuit _Carteſius_. </s>
  <s xml:id="echoid-s15244" xml:space="preserve">Eam <lb/>ego jam è linearum ſingulis appropriatarum deſcriptione conabor ali-<lb/>quatenus enucleatam dare; </s>
  <s xml:id="echoid-s15245" xml:space="preserve">qui ſanè modus rem præſertim elucidare <lb/>videtur, ac ob oculos ponere, agedum.</s>
  <s xml:id="echoid-s15246" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15247" xml:space="preserve">_Notetur_, In ſequentibus perpetim ad eaſdem ſeries redigi æquatio-<lb/>nes, quæ _coefficientes_ habent eaſdem.</s>
  <s xml:id="echoid-s15248" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div540" type="section" level="1" n="76">
<head xml:id="echoid-head79" style="it" xml:space="preserve">Æquationum<unsure/> Series prima.</head>
<p>
  <s xml:id="echoid-s15249" xml:space="preserve">_a_ + _b_ = _n_.</s>
  <s xml:id="echoid-s15250" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15251" xml:space="preserve">_aa_ + _ba_ = _nn_.</s>
  <s xml:id="echoid-s15252" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15253" xml:space="preserve">_a_<emph style="sub">3</emph> + _baa_ = _n_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15254" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15255" xml:space="preserve">_a_<emph style="sub">4</emph> + _ba_<emph style="sub">3</emph> = _n_<emph style="sub">4</emph>, &amp;</s>
  <s xml:id="echoid-s15256" xml:space="preserve">c.</s>
  <s xml:id="echoid-s15257" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15258" xml:space="preserve">Sumatur recta BA æqualis coefficienti _b_, &amp; </s>
  <s xml:id="echoid-s15259" xml:space="preserve">hæc verſus H indefini-<lb/>
<anchor type="note" xlink:label="note-0309-01a" xlink:href="note-0309-01"/>
tè protendatur; </s>
  <s xml:id="echoid-s15260" xml:space="preserve">ſint anguli RAH, SBH ſemirecti, ſintque lineæ <lb/>ALL, AMM, ANN tales, ut rectâ GK ductâ ad AH utcunque <lb/>perpendiculari (quæ dictas lineas ordine ſecet punctis L, M, N; </s>
  <s xml:id="echoid-s15261" xml:space="preserve">re-<lb/>ctaſque BS, AR punctis K, Z) ſit inter GZ, GK _media_ GL <anchor type="note" xlink:href="" symbol="*"/>, _bi-_
<anchor type="note" xlink:label="note-0309-02a" xlink:href="note-0309-02"/>
_media_ GM, _trimedia_ GM; </s>
  <s xml:id="echoid-s15262" xml:space="preserve">hæ lineæ propoſitarum æquationum <lb/>naturæ explicandæ inſervient. </s>
  <s xml:id="echoid-s15263" xml:space="preserve">Nam ſi AG (vel GZ) dicatur _a_; </s>
  <s xml:id="echoid-s15264" xml:space="preserve">erit <lb/>BG (vel GK) = _b_ + _a_; </s>
  <s xml:id="echoid-s15265" xml:space="preserve">atque GLq = _aa_ + _ba_; </s>
  <s xml:id="echoid-s15266" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15267" xml:space="preserve">GM cub. <lb/></s>
  <s xml:id="echoid-s15268" xml:space="preserve"> = _a_<emph style="sub">3</emph> + _baa_; </s>
  <s xml:id="echoid-s15269" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15270" xml:space="preserve">GN_qq_ = _a_<emph style="sub">4</emph> + _ba_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15271" xml:space="preserve"/>
</p>
<div xml:id="echoid-div540" type="float" level="2" n="1">
<note position="right" xlink:label="note-0309-01" xlink:href="note-0309-01a" xml:space="preserve">Fig. 206.</note>
<note symbol="*" position="right" xlink:label="note-0309-02" xlink:href="note-0309-02a" xml:space="preserve">Vid. pag. 90.</note>
</div>
<pb o="132" file="0310" n="325" rhead=""/>
</div>
<div xml:id="echoid-div542" type="section" level="1" n="77">
<head xml:id="echoid-head80" xml:space="preserve">_Notetur autem_,</head>
<p>
  <s xml:id="echoid-s15272" xml:space="preserve">1. </s>
  <s xml:id="echoid-s15273" xml:space="preserve">Ducta AD ad BH perpendiculari, ſi in hac capiatur AE = _n_; <lb/></s>
  <s xml:id="echoid-s15274" xml:space="preserve">ducatúrque EF ad AH parallela; </s>
  <s xml:id="echoid-s15275" xml:space="preserve">hujus cum lineis expoſitis interſe-<lb/>ctiones æquationum propoſitarum radices exhibebunt reſpectivè; </s>
  <s xml:id="echoid-s15276" xml:space="preserve">erit <lb/>
<anchor type="note" xlink:label="note-0310-01a" xlink:href="note-0310-01"/>
utique EK, vel EI, vel EM, vel EN æqualis ipſi _a_; <lb/></s>
  <s xml:id="echoid-s15277" xml:space="preserve">hoc eſt ipſis AG, concipiendo à ſingula interſectione deduci ad AH <lb/>perpendiculares, quæ puncta G determinet.</s>
  <s xml:id="echoid-s15278" xml:space="preserve"/>
</p>
<div xml:id="echoid-div542" type="float" level="2" n="1">
<note position="left" xlink:label="note-0310-01" xlink:href="note-0310-01a" xml:space="preserve">Fig. 206.</note>
</div>
<p>
  <s xml:id="echoid-s15279" xml:space="preserve">2. </s>
  <s xml:id="echoid-s15280" xml:space="preserve">Quò punctum G magîs à termino A removetur (&amp; </s>
  <s xml:id="echoid-s15281" xml:space="preserve">quidem poteſt <lb/>GA deſumi quavis deſignatâ major) eò ordinatæ GK, GL, GM, GN <lb/>magìs increſcunt; </s>
  <s xml:id="echoid-s15282" xml:space="preserve">adeo ut quantacunque ponatur AE, parallela EF <lb/>curvis occurſura ſit; </s>
  <s xml:id="echoid-s15283" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15284" xml:space="preserve">proinde ſemper habetur vera radix iſtarum <lb/>æquationum cuilibet conveniens; </s>
  <s xml:id="echoid-s15285" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15286" xml:space="preserve">ea tantùm una, quoniam EF <lb/>curvas iſtas unico puncto interſecat.</s>
  <s xml:id="echoid-s15287" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15288" xml:space="preserve">3. </s>
  <s xml:id="echoid-s15289" xml:space="preserve">_Curva_ ALL eſt _hyperbola æquilatera_, cujus _axis_ AB, reliquæ <lb/>AMM, ANN ſunt _hiperboliformes_.</s>
  <s xml:id="echoid-s15290" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15291" xml:space="preserve">4. </s>
  <s xml:id="echoid-s15292" xml:space="preserve">Si AO ſit {1/2} AB; </s>
  <s xml:id="echoid-s15293" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15294" xml:space="preserve">AP = {1/3} AB, &amp; </s>
  <s xml:id="echoid-s15295" xml:space="preserve">AQ = {1/4} AB, du-<lb/>cantúrque OT, PV, QX ad BS parallelæ, erunt hæ curvarum ALL, <lb/>AMM, ANN _aſymptoti_.</s>
  <s xml:id="echoid-s15296" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15297" xml:space="preserve">5. </s>
  <s xml:id="echoid-s15298" xml:space="preserve">Hinc conſtat in ſecundo gradu fore _a_ &amp;</s>
  <s xml:id="echoid-s15299" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s15300" xml:space="preserve">_n_ - {_b_/2}; </s>
  <s xml:id="echoid-s15301" xml:space="preserve">in tertio _a_&amp;</s>
  <s xml:id="echoid-s15302" xml:space="preserve">gt; <lb/></s>
  <s xml:id="echoid-s15303" xml:space="preserve">_n_ - {_b_/3}; </s>
  <s xml:id="echoid-s15304" xml:space="preserve">in quarto _a_&amp;</s>
  <s xml:id="echoid-s15305" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s15306" xml:space="preserve">_n_ - {_b_/4}; </s>
  <s xml:id="echoid-s15307" xml:space="preserve">quæ tamen inæqualitates, ſi AE <lb/>benemagna ſit, exiguæ erunt.</s>
  <s xml:id="echoid-s15308" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15309" xml:space="preserve">6. </s>
  <s xml:id="echoid-s15310" xml:space="preserve">Æquationibus iſtis nulla competit _maxima, vel minima_.</s>
  <s xml:id="echoid-s15311" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div544" type="section" level="1" n="78">
<head xml:id="echoid-head81" style="it" xml:space="preserve">Series ſecunda.</head>
<p>
  <s xml:id="echoid-s15312" xml:space="preserve">_a_ - _b_ = _n_.</s>
  <s xml:id="echoid-s15313" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15314" xml:space="preserve">_aa_ - _ba_ = _nn_.</s>
  <s xml:id="echoid-s15315" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15316" xml:space="preserve">_a_<emph style="sub">3</emph> - _baa_ = _n_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15317" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15318" xml:space="preserve">_a_<emph style="sub">4</emph> - _ba_<emph style="sub">3</emph> = _n_<emph style="sub">4</emph>, &amp;</s>
  <s xml:id="echoid-s15319" xml:space="preserve">c.</s>
  <s xml:id="echoid-s15320" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15321" xml:space="preserve">Sit rurſus AB = _b_; </s>
  <s xml:id="echoid-s15322" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15323" xml:space="preserve">indefinitè protrahatur AB verſus I, &amp; </s>
  <s xml:id="echoid-s15324" xml:space="preserve"><lb/>
<anchor type="note" xlink:label="note-0310-02a" xlink:href="note-0310-02"/>
ſint anguli RAI, SBI ſemirecti; </s>
  <s xml:id="echoid-s15325" xml:space="preserve">tum concipiantur curvæ BLL, <lb/>BMM, BNN tales, ut ſi utcunque ducatur GZ ad AI perpendicu-<lb/>laris (dictas lineas ſecans, utì cernis, punctis K, L, M, N, Z) ſit inter
<pb o="133" file="0311" n="326" rhead=""/>
GZ, GK media GL, bimedia GM, trimedia GN; </s>
  <s xml:id="echoid-s15326" xml:space="preserve">propoſitas æ-<lb/>quationes explicabunt hæ lineæ. </s>
  <s xml:id="echoid-s15327" xml:space="preserve">Nam ſi AG (vel GZ) vocetur _a_; <lb/></s>
  <s xml:id="echoid-s15328" xml:space="preserve">erit BG (vel GK) = _a_ - _b_; </s>
  <s xml:id="echoid-s15329" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15330" xml:space="preserve">GLq = _aa_ - _ba_; </s>
  <s xml:id="echoid-s15331" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15332" xml:space="preserve">GM cub. </s>
  <s xml:id="echoid-s15333" xml:space="preserve"><lb/> = _a_<emph style="sub">3</emph> - _baa_; </s>
  <s xml:id="echoid-s15334" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15335" xml:space="preserve">GN _qq_ = _a_<emph style="sub">4</emph> - _ba_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15336" xml:space="preserve"/>
</p>
<div xml:id="echoid-div544" type="float" level="2" n="1">
<note position="left" xlink:label="note-0310-02" xlink:href="note-0310-02a" xml:space="preserve">Fig. 207.</note>
</div>
</div>
<div xml:id="echoid-div546" type="section" level="1" n="79">
<head xml:id="echoid-head82" style="it" xml:space="preserve">Not.</head>
<p>
  <s xml:id="echoid-s15337" xml:space="preserve">1. </s>
  <s xml:id="echoid-s15338" xml:space="preserve">Ductâ AD ad AI perpendiculari, &amp; </s>
  <s xml:id="echoid-s15339" xml:space="preserve">EF ad AI parallelâ, ſi <lb/>AE ponatur æqualis ipſi _n_; </s>
  <s xml:id="echoid-s15340" xml:space="preserve">erunt EK, EL, EM, EN radices æqua-<lb/>tionum reſpectivæ, ſeu æquales quæſitis _a_.</s>
  <s xml:id="echoid-s15341" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15342" xml:space="preserve">2. </s>
  <s xml:id="echoid-s15343" xml:space="preserve">Quoniam ordinatæ GK, GL, GM, GN à termino B verſus I <lb/>infinitè excreſcunt, ſemper habetur una vera radix, &amp; </s>
  <s xml:id="echoid-s15344" xml:space="preserve">unica.</s>
  <s xml:id="echoid-s15345" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15346" xml:space="preserve">3. </s>
  <s xml:id="echoid-s15347" xml:space="preserve">Curva BLL eſt _hyperbola æquilatera_, cujus _axis_ AB, reliquæ <lb/>curvæ ſunt _hyperboliformes._</s>
  <s xml:id="echoid-s15348" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15349" xml:space="preserve">4. </s>
  <s xml:id="echoid-s15350" xml:space="preserve">Si AB biſecetur in O, triſecetur in P, quadriſecetur in Q, du-<lb/>cantúrque ad AR parallelæ OT, PV, QX, erunt hæ curvarum BLL, <lb/>BMM, BNN _aſymptoti._</s>
  <s xml:id="echoid-s15351" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15352" xml:space="preserve">5. </s>
  <s xml:id="echoid-s15353" xml:space="preserve">Hinc ſeqiutur in ſecundo gradu fore _a_ &amp;</s>
  <s xml:id="echoid-s15354" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s15355" xml:space="preserve">_n_ + {_b_/2}; </s>
  <s xml:id="echoid-s15356" xml:space="preserve">in tertio <lb/>_a_ &amp;</s>
  <s xml:id="echoid-s15357" xml:space="preserve">gt;_</s>
  <s xml:id="echoid-s15358" xml:space="preserve">n_ + {_b_/3}; </s>
  <s xml:id="echoid-s15359" xml:space="preserve">in quarto _a_ &amp;</s>
  <s xml:id="echoid-s15360" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s15361" xml:space="preserve">_n_ + {_b_/4}; </s>
  <s xml:id="echoid-s15362" xml:space="preserve">quòd ſi _n_ ſatis magna ſit, <lb/>iſtæ inæqualitates ad æqualitatem proximè accedunt.</s>
  <s xml:id="echoid-s15363" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15364" xml:space="preserve">6. </s>
  <s xml:id="echoid-s15365" xml:space="preserve">Verarum in his radicum habetur _minima;_ </s>
  <s xml:id="echoid-s15366" xml:space="preserve">ſcilicet ipſa AB, vel _b_.</s>
  <s xml:id="echoid-s15367" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div547" type="section" level="1" n="80">
<head xml:id="echoid-head83" style="it" xml:space="preserve">Series tertia.</head>
<p>
  <s xml:id="echoid-s15368" xml:space="preserve">_b_ - _a_ = _n_.</s>
  <s xml:id="echoid-s15369" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15370" xml:space="preserve">_ba_ - _aa_ = _nn_.</s>
  <s xml:id="echoid-s15371" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15372" xml:space="preserve">_baa_ - _a_<emph style="sub">3</emph> = _n_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15373" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15374" xml:space="preserve">_ba_<emph style="sub">3</emph>-_a_<emph style="sub">4</emph> = _n_<emph style="sub">4</emph>. </s>
  <s xml:id="echoid-s15375" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s15376" xml:space="preserve">c.</s>
  <s xml:id="echoid-s15377" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15378" xml:space="preserve">Sit AB = _b_, &amp; </s>
  <s xml:id="echoid-s15379" xml:space="preserve">anguli RAB, SBA ſemirecti; </s>
  <s xml:id="echoid-s15380" xml:space="preserve">tum curvæ <lb/>
<anchor type="note" xlink:label="note-0311-01a" xlink:href="note-0311-01"/>
ALB, AMB, ANB tales, ut ductâ rectâ GK ad AB utcunque <lb/>perpendiculari (quæ lineas expoſitas ſecet, ut vides) ſit inter AG <lb/>(ſeu GZ) &amp; </s>
  <s xml:id="echoid-s15381" xml:space="preserve">GK _media_ GL, _bimedia_ GM, _trimedia_ GN; </s>
  <s xml:id="echoid-s15382" xml:space="preserve">pro-<lb/>poſitas æquationes explicatas dabunt hæ lineæ. </s>
  <s xml:id="echoid-s15383" xml:space="preserve">Nam poſito fore AG <lb/> = _a_, erit GK = _b_ - _a_; </s>
  <s xml:id="echoid-s15384" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15385" xml:space="preserve">GLq = _ba_ - _aa_; </s>
  <s xml:id="echoid-s15386" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15387" xml:space="preserve">GMq = <lb/>_baa_ - _a_<emph style="sub">3</emph>. </s>
  <s xml:id="echoid-s15388" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15389" xml:space="preserve">GNq = _ba_<emph style="sub">3</emph> - _a_<emph style="sub">4</emph>.</s>
  <s xml:id="echoid-s15390" xml:space="preserve"/>
</p>
<div xml:id="echoid-div547" type="float" level="2" n="1">
<note position="right" xlink:label="note-0311-01" xlink:href="note-0311-01a" xml:space="preserve">Fig. 280.</note>
</div>
<pb o="134" file="0312" n="327" rhead=""/>
</div>
<div xml:id="echoid-div549" type="section" level="1" n="81">
<head xml:id="echoid-head84" style="it" xml:space="preserve">Not.</head>
<p>
  <s xml:id="echoid-s15391" xml:space="preserve">1. </s>
  <s xml:id="echoid-s15392" xml:space="preserve">Si in AD (ad ipſam AB perpendiculari) deſumatur AE = _n_; <lb/></s>
  <s xml:id="echoid-s15393" xml:space="preserve">
<anchor type="note" xlink:label="note-0312-01a" xlink:href="note-0312-01"/>
&amp; </s>
  <s xml:id="echoid-s15394" xml:space="preserve">ducatur EF ad AB parallela, hujuſce cum lineis expoſitis interſe-<lb/>ctiones exhibebunt radices _a_ reſpectivè.</s>
  <s xml:id="echoid-s15395" xml:space="preserve"/>
</p>
<div xml:id="echoid-div549" type="float" level="2" n="1">
<note position="left" xlink:label="note-0312-01" xlink:href="note-0312-01a" xml:space="preserve">Fig. 208.</note>
</div>
<p>
  <s xml:id="echoid-s15396" xml:space="preserve">2. </s>
  <s xml:id="echoid-s15397" xml:space="preserve">Cum ad haſce curvas ordinatæ ſemper terminatæ ſint, &amp; </s>
  <s xml:id="echoid-s15398" xml:space="preserve">inter <lb/>ipſas maxima quædam detur, hujus _ſeriei æquationes_, pro modulo aſ-<lb/>ſignatæ AE (vel _n_) ſubinde duas radices veras habent (cùm utique <lb/>fuerit AE curvæ maximâ ordinatâ minor reſpectivè, hoc eſt cùm EF <lb/>curvæ bis occurrerit) nonnunquam duntaxat unam (cum AE nempe <lb/>maximam adæquet, adeóque EF curvam contingat) aliquando nullam <lb/>(cum ſcilicet AE maximam excedat, adeoque nec EF curvæ unquam <lb/>occurrat).</s>
  <s xml:id="echoid-s15399" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15400" xml:space="preserve">3. </s>
  <s xml:id="echoid-s15401" xml:space="preserve">In ſecundo gradu ſi AO = OB, &amp; </s>
  <s xml:id="echoid-s15402" xml:space="preserve">ordinetur OT, erit OT <lb/>maxima; </s>
  <s xml:id="echoid-s15403" xml:space="preserve">(adeóque radicum una major quàm {AB/2}, altera minor) in <lb/>tertio, ſi AP = 2 PB, &amp; </s>
  <s xml:id="echoid-s15404" xml:space="preserve">ordinetur PV, erit PV maxima (unde <lb/>radicum una major erit quàm {1/3} AB, altera minor) demùm in quar-<lb/>to gradu ſi AQ = 3 QB, &amp; </s>
  <s xml:id="echoid-s15405" xml:space="preserve">ordinetur QX, erit QX _maxima_ <lb/>(&amp; </s>
  <s xml:id="echoid-s15406" xml:space="preserve">hinc una radicum ſemper major, quàm {1/4} AB, &amp; </s>
  <s xml:id="echoid-s15407" xml:space="preserve">altera minor).</s>
  <s xml:id="echoid-s15408" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15409" xml:space="preserve">4. </s>
  <s xml:id="echoid-s15410" xml:space="preserve">Hinc conſectatur, ſi fuerit, in ſecundo gradu n &amp;</s>
  <s xml:id="echoid-s15411" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s15412" xml:space="preserve">{_b_/2}; </s>
  <s xml:id="echoid-s15413" xml:space="preserve">in tertio <lb/>_n_<emph style="sub">2</emph>&amp;</s>
  <s xml:id="echoid-s15414" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s15415" xml:space="preserve">{4_b_<emph style="sub">3</emph>/9} - {8_b_<emph style="sub">3</emph>/27<unsure/>} = {4 _b_<emph style="sub">3</emph>/27}; </s>
  <s xml:id="echoid-s15416" xml:space="preserve">in quarto _n_<emph style="sub">4</emph>&amp;</s>
  <s xml:id="echoid-s15417" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s15418" xml:space="preserve">{27/64}_b_<emph style="sub">4</emph> - {81/256}_b_<emph style="sub">4</emph> = <lb/>{27_b_<emph style="sub">4</emph>/256}; </s>
  <s xml:id="echoid-s15419" xml:space="preserve">nullam dari radicem.</s>
  <s xml:id="echoid-s15420" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15421" xml:space="preserve">5. </s>
  <s xml:id="echoid-s15422" xml:space="preserve">Omnium radicum _maxima_ eſt ipſa AB, vel _b_.</s>
  <s xml:id="echoid-s15423" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15424" xml:space="preserve">6. </s>
  <s xml:id="echoid-s15425" xml:space="preserve">Omnium curvarum communis _interſectio_ (ſeu _nodus_) eſt pun-<lb/>ctum T; </s>
  <s xml:id="echoid-s15426" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15427" xml:space="preserve">ſi fuerit _n_ = {_b_/2}; </s>
  <s xml:id="echoid-s15428" xml:space="preserve">ſemper AO (vel {_b_/2}) eſt una radix.</s>
  <s xml:id="echoid-s15429" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15430" xml:space="preserve">7. </s>
  <s xml:id="echoid-s15431" xml:space="preserve">Curva ALB eſt _Circulus_, reliquæ AMB, ANB eum quo-<lb/>dammodo referunt.</s>
  <s xml:id="echoid-s15432" xml:space="preserve"/>
</p>
<note position="right" xml:space="preserve"> <lb/>1. # 2. # 3. <lb/>_a_ + _b_ = _n_ \\ _a_ + _b_ = {_nn_/_a_} \\ _a_ + _b_ = {_n_<emph style="sub">3</emph>/_aa_} \\ _a_ + _b_ = {_n_4<emph style="sub">4</emph>/_a_<emph style="sub">3</emph>} # _a_ - _b_ = _n_. \\ _a_ - _b_ = {_nn_/_a_} \\ _a_ - _b_ = {_n_<emph style="sub">3</emph>/_aa_} \\ _a_ - _b_ = {_n_<emph style="sub">4</emph>/_a_<emph style="sub">3</emph>3} # {_b_ - _a_ = _n_. \\ _b_ - _a_ = {_nn_/_a_} \\ _b_ - _a_ = {_n_<emph style="sub">3</emph>/aa} \\ _b_ - _a_ = {_n_<emph style="sub">4</emph>/_a_<emph style="sub">3</emph>} <lb/></note>
<pb o="135" file="0313" n="328" rhead=""/>
<p>
  <s xml:id="echoid-s15433" xml:space="preserve">Aliter (&amp; </s>
  <s xml:id="echoid-s15434" xml:space="preserve">forte commodiùs; </s>
  <s xml:id="echoid-s15435" xml:space="preserve">pro ſingulo trium ſerierum gradu tan-<lb/>tùm unam adhibendo lineam) explicantur iſtæ præcedaneæ æquatio-<lb/>nes, hoc pacto:</s>
  <s xml:id="echoid-s15436" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15437" xml:space="preserve">Sit AH recta indefinitè protenſa, &amp; </s>
  <s xml:id="echoid-s15438" xml:space="preserve">huic perpendicularis AD; </s>
  <s xml:id="echoid-s15439" xml:space="preserve">in <lb/>
<anchor type="note" xlink:label="note-0313-01a" xlink:href="note-0313-01"/>
qua ſumatur AB = _n_, &amp; </s>
  <s xml:id="echoid-s15440" xml:space="preserve">ducatur BK ad AH parallela, tum ſint <lb/>lineæ LXL, MXM, NXN tales<unsure/>, ut ſumpto in AH quocunque <lb/>puncto G, &amp; </s>
  <s xml:id="echoid-s15441" xml:space="preserve">ductâ GK ad AD parallelâ, ſit in proportione AG ad <lb/>GK (vel AB) proportione _tertia_ GL, _quarta_ GM, _quinta_ GN; </s>
  <s xml:id="echoid-s15442" xml:space="preserve">hæ <lb/>lineæ propoſitarum æquationum naturæ explicandæ inſervient.</s>
  <s xml:id="echoid-s15443" xml:space="preserve"/>
</p>
<div xml:id="echoid-div550" type="float" level="2" n="2">
<note position="right" xlink:label="note-0313-01" xlink:href="note-0313-01a" xml:space="preserve">Fig. 209, <lb/>210.</note>
</div>
<p>
  <s xml:id="echoid-s15444" xml:space="preserve">Nam ſumpta AE = _b_ (ſumatur autem AE ob primam ſeriem <lb/>ad partes I, ob ſecundam &amp; </s>
  <s xml:id="echoid-s15445" xml:space="preserve">tertiam ad partes H) &amp; </s>
  <s xml:id="echoid-s15446" xml:space="preserve">fiat an-<lb/>gulus FEH ſemirectus (iſte quidem pro prima &amp; </s>
  <s xml:id="echoid-s15447" xml:space="preserve">ſecunda ſe-<lb/>rie inclinans verſus H, pro tertia reclinans ab H, ut Schema ſatis <lb/>monſtrat) tum rectæ EF cum expoſitis lineis interſectiones reſpectivæ <lb/>radices a determinabunt; </s>
  <s xml:id="echoid-s15448" xml:space="preserve">nempe ſi per has ductæ concipiantur ad AH <lb/>perpendiculares(LG, MG, NG) erunt interceptæ AG radicibus _a_ <lb/>æquales reſpectivè.</s>
  <s xml:id="echoid-s15449" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div552" type="section" level="1" n="82">
<head xml:id="echoid-head85" style="it" xml:space="preserve">Not.</head>
<p>
  <s xml:id="echoid-s15450" xml:space="preserve">Exhinc conſtat, quòd</s>
</p>
<p>
  <s xml:id="echoid-s15451" xml:space="preserve">1. </s>
  <s xml:id="echoid-s15452" xml:space="preserve">In hac explicatione _coefficiens b_ indeterminata habetur; </s>
  <s xml:id="echoid-s15453" xml:space="preserve">ut in præ-<lb/>cedentibus ipſa _n_.</s>
  <s xml:id="echoid-s15454" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15455" xml:space="preserve">2. </s>
  <s xml:id="echoid-s15456" xml:space="preserve">In prima &amp; </s>
  <s xml:id="echoid-s15457" xml:space="preserve">ſecunda ſerie ſemper una poſitiva radix habetur, &amp; </s>
  <s xml:id="echoid-s15458" xml:space="preserve"><lb/>unica.</s>
  <s xml:id="echoid-s15459" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15460" xml:space="preserve">3. </s>
  <s xml:id="echoid-s15461" xml:space="preserve">In ſecunda ſerie minima radix ipſi AB, vel _n_ æquatur.</s>
  <s xml:id="echoid-s15462" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15463" xml:space="preserve">4. </s>
  <s xml:id="echoid-s15464" xml:space="preserve">Communis omnium linearum _nodus_ eſt _punctum_ X, ubi BX <lb/>(vel _a_) = _n_.</s>
  <s xml:id="echoid-s15465" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15466" xml:space="preserve">5. </s>
  <s xml:id="echoid-s15467" xml:space="preserve">In tertia ſerie ſubindè duæ habentur radices poſitivæ (quando <lb/>ſcilicet EF curvas bis ſecat) nonnunquam una tantùm (cùm EF ip-<lb/>ſarum aliquam contingat; </s>
  <s xml:id="echoid-s15468" xml:space="preserve">id quod accidit in ſecundo gradu cùm <lb/>a = {_b_/2}; </s>
  <s xml:id="echoid-s15469" xml:space="preserve">in tertio cùm a = {2/3}_b_; </s>
  <s xml:id="echoid-s15470" xml:space="preserve">in quarto cùm a = {3/4}_b_) aliquando <lb/>nulla, cùm EF infra tangentes cadit, &amp; </s>
  <s xml:id="echoid-s15471" xml:space="preserve">adeò nuſquam curvis occur-<lb/>rit.</s>
  <s xml:id="echoid-s15472" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15473" xml:space="preserve">6. </s>
  <s xml:id="echoid-s15474" xml:space="preserve">Secundi gradûs curva eſt _hyperbola_, reliquæ _hyperloliformes_, <lb/>quarum communes _aſymptoti_ ſunt rectæ AH, AD.</s>
  <s xml:id="echoid-s15475" xml:space="preserve"/>
</p>
<pb o="136" file="0314" n="329" rhead=""/>
</div>
<div xml:id="echoid-div553" type="section" level="1" n="83">
<head xml:id="echoid-head86" style="it" xml:space="preserve">Series quarta.</head>
<p>
  <s xml:id="echoid-s15476" xml:space="preserve">_a_ + {_cc_/_a_} = _n_.</s>
  <s xml:id="echoid-s15477" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15478" xml:space="preserve">_aa_ + _cc_ = _nn_.</s>
  <s xml:id="echoid-s15479" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15480" xml:space="preserve">_a_<emph style="sub">3</emph> + _cca_ = _n_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15481" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15482" xml:space="preserve">_a_<emph style="sub">4</emph> + _ccaa_ = _n_<emph style="sub">4</emph>.</s>
  <s xml:id="echoid-s15483" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15484" xml:space="preserve">Sit recta indefinitè protenſa AH, &amp; </s>
  <s xml:id="echoid-s15485" xml:space="preserve">huic perpendicularis AD; <lb/></s>
  <s xml:id="echoid-s15486" xml:space="preserve">fiat autem angulus RAH ſemirectus; </s>
  <s xml:id="echoid-s15487" xml:space="preserve">tum utcunque ducatur GZK <lb/>
<anchor type="note" xlink:label="note-0314-01a" xlink:href="note-0314-01"/>
ad AD parallela; </s>
  <s xml:id="echoid-s15488" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15489" xml:space="preserve">facto AG. </s>
  <s xml:id="echoid-s15490" xml:space="preserve">AG:</s>
  <s xml:id="echoid-s15491" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s15492" xml:space="preserve">ZK; </s>
  <s xml:id="echoid-s15493" xml:space="preserve">per Kintra <lb/>angulum DAR deſcribatur _hyperbola_ KXK; </s>
  <s xml:id="echoid-s15494" xml:space="preserve">ſint denuò curvæ CLL, <lb/>AMM, ANN tales, ut inter GZ, GK ſint _media_ GL, _bimedia_ <lb/>GM, _trimedia_ GN; </s>
  <s xml:id="echoid-s15495" xml:space="preserve">hæ propoſito deſervient. </s>
  <s xml:id="echoid-s15496" xml:space="preserve">Nam ſi AG (vel <lb/>GZ) dicatur _a_, erit GK = _a_ + {_cc_/_a_}; </s>
  <s xml:id="echoid-s15497" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15498" xml:space="preserve">GLq = _aa_ + _cc_; </s>
  <s xml:id="echoid-s15499" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15500" xml:space="preserve"><lb/>GMcub = _a_<emph style="sub">3</emph> + _cca_; </s>
  <s xml:id="echoid-s15501" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15502" xml:space="preserve">GNqq = _a_<emph style="sub">4</emph> + _ccaa_.</s>
  <s xml:id="echoid-s15503" xml:space="preserve"/>
</p>
<div xml:id="echoid-div553" type="float" level="2" n="1">
<note position="left" xlink:label="note-0314-01" xlink:href="note-0314-01a" xml:space="preserve">Fig. 211.</note>
</div>
</div>
<div xml:id="echoid-div555" type="section" level="1" n="84">
<head xml:id="echoid-head87" style="it" xml:space="preserve">Not.</head>
<p>
  <s xml:id="echoid-s15504" xml:space="preserve">1. </s>
  <s xml:id="echoid-s15505" xml:space="preserve">Deſignantur radices, ut in præcedentibus, poſitâ AE = _n_, &amp; </s>
  <s xml:id="echoid-s15506" xml:space="preserve">ductâ <lb/>EF ad AH parallelâ.</s>
  <s xml:id="echoid-s15507" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15508" xml:space="preserve">2. </s>
  <s xml:id="echoid-s15509" xml:space="preserve">Si AP = AC, erit PX ad _hyperbolam_ KXK ordinatarum _mi_-<lb/>_nima_; </s>
  <s xml:id="echoid-s15510" xml:space="preserve">unde ſi AE (vel _n_) &amp;</s>
  <s xml:id="echoid-s15511" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s15512" xml:space="preserve">PX; </s>
  <s xml:id="echoid-s15513" xml:space="preserve">nulla dabitur radix in primo <lb/>gradu.</s>
  <s xml:id="echoid-s15514" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15515" xml:space="preserve">3. </s>
  <s xml:id="echoid-s15516" xml:space="preserve">Curva CLL eſt _hyperbola æquilatera_, cujus _centrum_ A, _ſemi_-<lb/>_axis_ AC; </s>
  <s xml:id="echoid-s15517" xml:space="preserve">quæ &amp; </s>
  <s xml:id="echoid-s15518" xml:space="preserve">ordinatarum eſt _minima_; </s>
  <s xml:id="echoid-s15519" xml:space="preserve">alioquin ſi _n_&amp;</s>
  <s xml:id="echoid-s15520" xml:space="preserve">gt;_</s>
  <s xml:id="echoid-s15521" xml:space="preserve">c_, ſem-<lb/>per una vera radix habetur, &amp; </s>
  <s xml:id="echoid-s15522" xml:space="preserve">unica.</s>
  <s xml:id="echoid-s15523" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15524" xml:space="preserve">4. </s>
  <s xml:id="echoid-s15525" xml:space="preserve">Reliquæ AMM, ANN ſunt hyperboliformes ad infinitum <lb/>excurrentes; </s>
  <s xml:id="echoid-s15526" xml:space="preserve">unde ſemper una vera radix habetur, neque plures.</s>
  <s xml:id="echoid-s15527" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15528" xml:space="preserve">5. </s>
  <s xml:id="echoid-s15529" xml:space="preserve">Si fuerit Y α = {1/2} YX; </s>
  <s xml:id="echoid-s15530" xml:space="preserve">Y β = {1/3}YX; </s>
  <s xml:id="echoid-s15531" xml:space="preserve">Y γ = {1/4} YX, &amp; </s>
  <s xml:id="echoid-s15532" xml:space="preserve">per <lb/>puncta α, β γ, traductæ concipiantur _hpperbola<unsure/>_ (habentes &amp; </s>
  <s xml:id="echoid-s15533" xml:space="preserve">ipſæ _a_-<lb/>_ſymptotos_ DA, AR) α λ, β μ, γ ν; </s>
  <s xml:id="echoid-s15534" xml:space="preserve">erunt hæ ipſarum curvarum <lb/>CLL, AMM, ANN _aſymptoti_. </s>
  <s xml:id="echoid-s15535" xml:space="preserve">(Similes etiam _aſymptoti_ con-<lb/>veniunt lineis poſthac deſcribendis, quanquam de illis conticeamus.)</s>
  <s xml:id="echoid-s15536" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15537" xml:space="preserve">6. </s>
  <s xml:id="echoid-s15538" xml:space="preserve">Hinc in ſecundo gradu _a_ + {_cc_/2_a_}&amp;</s>
  <s xml:id="echoid-s15539" xml:space="preserve">gt;_</s>
  <s xml:id="echoid-s15540" xml:space="preserve">n_; </s>
  <s xml:id="echoid-s15541" xml:space="preserve">in tertio _a_ + {_cc_/3_a_}&amp;</s>
  <s xml:id="echoid-s15542" xml:space="preserve">gt;_</s>
  <s xml:id="echoid-s15543" xml:space="preserve">n_;</s>
  <s xml:id="echoid-s15544" xml:space="preserve">
<pb o="137" file="0315" n="330" rhead=""/>
in quarto _a_ + {_cc_/4_a_}&amp;</s>
  <s xml:id="echoid-s15545" xml:space="preserve">gt;_</s>
  <s xml:id="echoid-s15546" xml:space="preserve">n_; </s>
  <s xml:id="echoid-s15547" xml:space="preserve">quæ tamen inæqualitas eo minor eſt, quò <lb/>AE (vel _n_) major exiſtit.</s>
  <s xml:id="echoid-s15548" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15549" xml:space="preserve">_a_ + {_cc_/_a_} = _n_.</s>
  <s xml:id="echoid-s15550" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15551" xml:space="preserve">_a_ + {_cc_/_a_} = {_nn_/_a_}.</s>
  <s xml:id="echoid-s15552" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15553" xml:space="preserve">_a_ + {_cc_/_a_} = {_n_<emph style="sub">3</emph>/_aa_}.</s>
  <s xml:id="echoid-s15554" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15555" xml:space="preserve">_a_ + {_cc_/_a_} = {_n_<emph style="sub">4</emph>/_a_<emph style="sub">3</emph>}.</s>
  <s xml:id="echoid-s15556" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15557" xml:space="preserve">Poſſit hæc ſeries explicari juxta præcedentium modum ſecundum, <lb/>
<anchor type="note" xlink:label="note-0315-01a" xlink:href="note-0315-01"/>
&amp; </s>
  <s xml:id="echoid-s15558" xml:space="preserve">eaſdem adhibendo curvas LXL, MXM, NXN; </s>
  <s xml:id="echoid-s15559" xml:space="preserve">quarum nimi-<lb/>rum proprietas eſt, ut rectâ GK ductâ ad AH utcunque perpendicu-<lb/>lari, ſit GL = {_nn_/AG}; </s>
  <s xml:id="echoid-s15560" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15561" xml:space="preserve">GM = {_n_<emph style="sub">3</emph>/AGq}; </s>
  <s xml:id="echoid-s15562" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15563" xml:space="preserve">GN = {_n_<emph style="sub">4</emph>/AGcub}.</s>
  <s xml:id="echoid-s15564" xml:space="preserve"/>
</p>
<div xml:id="echoid-div555" type="float" level="2" n="1">
<note position="right" xlink:label="note-0315-01" xlink:href="note-0315-01a" xml:space="preserve">Fig. 212.</note>
</div>
<p>
  <s xml:id="echoid-s15565" xml:space="preserve">Nam ſi fiat angulus HAR ſemirectus, &amp; </s>
  <s xml:id="echoid-s15566" xml:space="preserve">utcunque ducatur GEO <lb/>ad AH perpendicularis; </s>
  <s xml:id="echoid-s15567" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15568" xml:space="preserve">ſit GE. </s>
  <s xml:id="echoid-s15569" xml:space="preserve">_c_:</s>
  <s xml:id="echoid-s15570" xml:space="preserve">:_c_. </s>
  <s xml:id="echoid-s15571" xml:space="preserve">EO; </s>
  <s xml:id="echoid-s15572" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15573" xml:space="preserve">per O intra a-<lb/>ſymptotos AD, AR deſcribatur _hyperbola_ OO; </s>
  <s xml:id="echoid-s15574" xml:space="preserve">hujuſce cum expo-<lb/>ſitis lineis LXL, MXM, NXN interſectiones, radices _a_ reſpectivas <lb/>determinabunt; </s>
  <s xml:id="echoid-s15575" xml:space="preserve">ductis utique LG, MG, NG ad AH perpendicu-<lb/>laribus; </s>
  <s xml:id="echoid-s15576" xml:space="preserve">erunt interceptæ AG ipſis _a_ æquales reſpectivè.</s>
  <s xml:id="echoid-s15577" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15578" xml:space="preserve">Poſſint conſimili modo ſubſequentes omnes æquationes explicari; <lb/></s>
  <s xml:id="echoid-s15579" xml:space="preserve">ſed eas modo duntaxat priore dabimus expoſitas.</s>
  <s xml:id="echoid-s15580" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div557" type="section" level="1" n="85">
<head xml:id="echoid-head88" style="it" xml:space="preserve">Series quinta.</head>
<note position="right" xml:space="preserve">Fig. 213.</note>
<p>
  <s xml:id="echoid-s15581" xml:space="preserve">{_cc_/_a_} - _a_ = _n_.</s>
  <s xml:id="echoid-s15582" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15583" xml:space="preserve">_cc_ - _aa_ = _nn_.</s>
  <s xml:id="echoid-s15584" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15585" xml:space="preserve">_cca_ - _a_<emph style="sub">3</emph> = _n_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15586" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15587" xml:space="preserve">_ccaa_ - _a_<emph style="sub">4</emph> = _n_<emph style="sub">4</emph>.</s>
  <s xml:id="echoid-s15588" xml:space="preserve"/>
</p>
<pb o="138" file="0316" n="331" rhead=""/>
</div>
<div xml:id="echoid-div558" type="section" level="1" n="86">
<head xml:id="echoid-head89" style="it" xml:space="preserve">Series ſexta.</head>
<p>
  <s xml:id="echoid-s15589" xml:space="preserve">_a_ - {_cc_/_a_} = _x_.</s>
  <s xml:id="echoid-s15590" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15591" xml:space="preserve">_aa_ - _cc_ = _nn_.</s>
  <s xml:id="echoid-s15592" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15593" xml:space="preserve">_a_<emph style="sub">3</emph> - _cca_ = _n_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15594" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15595" xml:space="preserve">_a_<emph style="sub">4</emph> - _ccaa_ = _n_<emph style="sub">4</emph>.</s>
  <s xml:id="echoid-s15596" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15597" xml:space="preserve">Fiat angulus RAI ſemirectus, &amp; </s>
  <s xml:id="echoid-s15598" xml:space="preserve">AD ad AI perpendicularis; <lb/></s>
  <s xml:id="echoid-s15599" xml:space="preserve">
<anchor type="note" xlink:label="note-0316-01a" xlink:href="note-0316-01"/>
in qua AC = _c_; </s>
  <s xml:id="echoid-s15600" xml:space="preserve">tum utcunque ductâ GZ ad AD parallelâ, ſit <lb/>AG (vel GZ). </s>
  <s xml:id="echoid-s15601" xml:space="preserve">AC:</s>
  <s xml:id="echoid-s15602" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s15603" xml:space="preserve">ZK, &amp; </s>
  <s xml:id="echoid-s15604" xml:space="preserve">per K, intra angulum DAR <lb/>deſcribatur _hyperbola_ KYK; </s>
  <s xml:id="echoid-s15605" xml:space="preserve">tum ſint curvæ CLYHLλ, AMYHMμ, <lb/>ANYHN ν<unsure/> tales, ut inter AG (vel GZ) &amp; </s>
  <s xml:id="echoid-s15606" xml:space="preserve">GK ſit _media_ GL, <lb/>_bimedia_ GM, _trimedia_ GN; </s>
  <s xml:id="echoid-s15607" xml:space="preserve">hæ propofito deſervient.</s>
  <s xml:id="echoid-s15608" xml:space="preserve"/>
</p>
<div xml:id="echoid-div558" type="float" level="2" n="1">
<note position="left" xlink:label="note-0316-01" xlink:href="note-0316-01a" xml:space="preserve">Fig. 213</note>
</div>
<p>
  <s xml:id="echoid-s15609" xml:space="preserve">Conſtat hoc, ut in præcedente; </s>
  <s xml:id="echoid-s15610" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15611" xml:space="preserve">quo pacto radices reſpectivè <lb/>determinantur. </s>
  <s xml:id="echoid-s15612" xml:space="preserve">Verùm adnotetur prætereà.</s>
  <s xml:id="echoid-s15613" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div560" type="section" level="1" n="87">
<head xml:id="echoid-head90" style="it" xml:space="preserve">Not.</head>
<p>
  <s xml:id="echoid-s15614" xml:space="preserve">1. </s>
  <s xml:id="echoid-s15615" xml:space="preserve">Curvæ CLH, AMH, ANH ad quintam ſeriem pertinent; </s>
  <s xml:id="echoid-s15616" xml:space="preserve">re-<lb/>liquæ HL λ, HM μ, HN ν ad ſextam.</s>
  <s xml:id="echoid-s15617" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15618" xml:space="preserve">2. </s>
  <s xml:id="echoid-s15619" xml:space="preserve">Quoad curvas ad quintam ſeriem pertinentes; </s>
  <s xml:id="echoid-s15620" xml:space="preserve">ſi A φ = √{ACq/2}; <lb/></s>
  <s xml:id="echoid-s15621" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15622" xml:space="preserve">ordinetur φ Y; </s>
  <s xml:id="echoid-s15623" xml:space="preserve">erit Y communis linearum interſectio, ſeu _no_-<lb/>_dus._</s>
  <s xml:id="echoid-s15624" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15625" xml:space="preserve">3. </s>
  <s xml:id="echoid-s15626" xml:space="preserve">In harum primo gradu ordinata AK eſt inſinita in ſecundo AC <lb/>eſt maxima; </s>
  <s xml:id="echoid-s15627" xml:space="preserve">in tertio ſi fuerit AP = √{ACq/3}, &amp; </s>
  <s xml:id="echoid-s15628" xml:space="preserve">ordinetur PV, <lb/>erit PV maxima(unde radicum una ſemper major eſt quam √{ACq/3} <lb/>altera minor) in quarto ſi AQ = √{ACq/4} = {AC/2}, &amp; </s>
  <s xml:id="echoid-s15629" xml:space="preserve">ordinetur QX, <lb/>erit QX maxima (unde radicum una major erit, altera minor ipsâ <lb/>{AC/2}).</s>
  <s xml:id="echoid-s15630" xml:space="preserve"/>
</p>
<pb o="139" file="0317" n="332" rhead=""/>
<p>
  <s xml:id="echoid-s15631" xml:space="preserve">4. </s>
  <s xml:id="echoid-s15632" xml:space="preserve">Conſequentèr in harum ſecundo gradu ſin &amp;</s>
  <s xml:id="echoid-s15633" xml:space="preserve">gt;_</s>
  <s xml:id="echoid-s15634" xml:space="preserve">c_; </s>
  <s xml:id="echoid-s15635" xml:space="preserve">in tertio, ſi _n_<emph style="sub">3</emph> <lb/>&amp;</s>
  <s xml:id="echoid-s15636" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s15637" xml:space="preserve">_cc_√{_cc_/3} - {_cc_/3} √ {_cc_/3} = {2/3}_cc_ √ {_cc_/3}; </s>
  <s xml:id="echoid-s15638" xml:space="preserve">vel _n_<emph style="sub">6</emph>&amp;</s>
  <s xml:id="echoid-s15639" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s15640" xml:space="preserve">{@@/27}_c_<unsure/><emph style="sub">6</emph>; </s>
  <s xml:id="echoid-s15641" xml:space="preserve">in quar-<lb/>to ſi _n_<emph style="sub">4</emph>&amp;</s>
  <s xml:id="echoid-s15642" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s15643" xml:space="preserve">{_c_<emph style="sub">4</emph>/4} - {_c_<emph style="sub">4</emph>/16} = {3/16}_c_<emph style="sub">4</emph>; </s>
  <s xml:id="echoid-s15644" xml:space="preserve">nulla radix habetur; </s>
  <s xml:id="echoid-s15645" xml:space="preserve">unam in iſtis <lb/>caſibus recta EF curvas ſupergreditur; </s>
  <s xml:id="echoid-s15646" xml:space="preserve">nec iis occurrit.</s>
  <s xml:id="echoid-s15647" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15648" xml:space="preserve">5. </s>
  <s xml:id="echoid-s15649" xml:space="preserve">Itidem in his omnibus maxima poſſibilis radix eſt AH = AC.</s>
  <s xml:id="echoid-s15650" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15651" xml:space="preserve">6. </s>
  <s xml:id="echoid-s15652" xml:space="preserve">Curva CYH eſt _Circuli quadrans_, reliquæ AMH, ANH <lb/>quodammodo κυχλο{ει}δ{ετ}ς.</s>
  <s xml:id="echoid-s15653" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15654" xml:space="preserve">7. </s>
  <s xml:id="echoid-s15655" xml:space="preserve">Ad ſextam ſeriem pertinentium curva HLL eſt _byperbola æqui_-<lb/>_latera_, cujus axis AH; </s>
  <s xml:id="echoid-s15656" xml:space="preserve">reliquæ ſunt _Hyperboliformes_. </s>
  <s xml:id="echoid-s15657" xml:space="preserve">Unde quoad <lb/>hanc ſeriem liquent cætera.</s>
  <s xml:id="echoid-s15658" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div561" type="section" level="1" n="88">
<head xml:id="echoid-head91" style="it" xml:space="preserve">Series ſeptima.</head>
<p>
  <s xml:id="echoid-s15659" xml:space="preserve">_a_ + _b_ + {_cc_/_a_} = _n_.</s>
  <s xml:id="echoid-s15660" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15661" xml:space="preserve">_aa_ + _ba_ + _cc_ = _nn._</s>
  <s xml:id="echoid-s15662" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15663" xml:space="preserve">_a_<emph style="sub">3</emph> + _baa_ + _cca_ = _n_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15664" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15665" xml:space="preserve">_a_<emph style="sub">4</emph> + _ba_<emph style="sub">3</emph> + _ccaa_ = _n_<emph style="sub">4</emph>, &amp;</s>
  <s xml:id="echoid-s15666" xml:space="preserve">c<unsure/>.</s>
  <s xml:id="echoid-s15667" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15668" xml:space="preserve">In recta BAH indefinitè protensâ capiatur AB = _b_; </s>
  <s xml:id="echoid-s15669" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15670" xml:space="preserve">in AD <lb/>
<anchor type="note" xlink:label="note-0317-01a" xlink:href="note-0317-01"/>
ad BH perpendiculari ſit AC = _c_; </s>
  <s xml:id="echoid-s15671" xml:space="preserve">ſint etiam anguli HAR, HBS Semi-<lb/>recti; </s>
  <s xml:id="echoid-s15672" xml:space="preserve">tum arbitrariè ductâ GY ad AH perpendiculari quæ ipſam <lb/>BS ſecet in Y; </s>
  <s xml:id="echoid-s15673" xml:space="preserve">fiat AG. </s>
  <s xml:id="echoid-s15674" xml:space="preserve">AC:</s>
  <s xml:id="echoid-s15675" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s15676" xml:space="preserve">YK; </s>
  <s xml:id="echoid-s15677" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15678" xml:space="preserve">per K intra angulum <lb/>DVS deſcribatur _hyperbola_ KKK; </s>
  <s xml:id="echoid-s15679" xml:space="preserve">ſint demum curvæ CLL, AMM, <lb/>ANN tales, ut inter AG (vel GZ) &amp; </s>
  <s xml:id="echoid-s15680" xml:space="preserve">GK ſit _media_ GL, _bime_-<lb/>_dia_ GM, _trimedia_ GN; </s>
  <s xml:id="echoid-s15681" xml:space="preserve">hæ ſatisfacient negotio. </s>
  <s xml:id="echoid-s15682" xml:space="preserve">Nam eſt GK = _a_ <lb/>+ _b_ + {_cc_/_a_}; </s>
  <s xml:id="echoid-s15683" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15684" xml:space="preserve">GLq = _aa_ + _ba_ + _cc_; </s>
  <s xml:id="echoid-s15685" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15686" xml:space="preserve">GMcub = _a_<emph style="sub">3</emph> + _baa_ <lb/>+ _cca_; </s>
  <s xml:id="echoid-s15687" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15688" xml:space="preserve">GNqq = _a_<emph style="sub">4</emph> + _ba_<emph style="sub">3</emph> + _ccaa_.</s>
  <s xml:id="echoid-s15689" xml:space="preserve"/>
</p>
<div xml:id="echoid-div561" type="float" level="2" n="1">
<note position="right" xlink:label="note-0317-01" xlink:href="note-0317-01a" xml:space="preserve">Fig. 214.</note>
</div>
</div>
<div xml:id="echoid-div563" type="section" level="1" n="89">
<head xml:id="echoid-head92" style="it" xml:space="preserve">Not.</head>
<p>
  <s xml:id="echoid-s15690" xml:space="preserve">1. </s>
  <s xml:id="echoid-s15691" xml:space="preserve">Secundi gradûs curva CLL eſt pars _hyperbolæ æquilateræ_, cujus <lb/>_centrum_ O, ipſam AB biſecans; </s>
  <s xml:id="echoid-s15692" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15693" xml:space="preserve">ſiquidem AC&amp;</s>
  <s xml:id="echoid-s15694" xml:space="preserve">gt;</s>
  <s xml:id="echoid-s15695" xml:space="preserve">AO, eſt OH <lb/>(ad AB perpendicularis, &amp;)</s>
  <s xml:id="echoid-s15696" xml:space="preserve"> = √ ACq - AO qejus _ſemiaxis_; <lb/></s>
  <s xml:id="echoid-s15697" xml:space="preserve">ſin AC&amp;</s>
  <s xml:id="echoid-s15698" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s15699" xml:space="preserve">AO, ejus axis eſt OI = √ AOq - ACq. </s>
  <s xml:id="echoid-s15700" xml:space="preserve">reliquæ <lb/>verò curvæ AMM, ANN ſunt _hyperboliformes_.</s>
  <s xml:id="echoid-s15701" xml:space="preserve"/>
</p>
<pb o="140" file="0318" n="333" rhead=""/>
<p>
  <s xml:id="echoid-s15702" xml:space="preserve">2. </s>
  <s xml:id="echoid-s15703" xml:space="preserve">Hinc conſtat in ſecundo gradu ſi fuerit _n_&amp;</s>
  <s xml:id="echoid-s15704" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s15705" xml:space="preserve">C, nullam veram <lb/>radicem dari; </s>
  <s xml:id="echoid-s15706" xml:space="preserve">alioquin in omnibus una ſemper habetur, &amp; </s>
  <s xml:id="echoid-s15707" xml:space="preserve">unica; <lb/></s>
  <s xml:id="echoid-s15708" xml:space="preserve">quoniam recta EF curvas ſemel interſecabit, nec pluries,</s>
</p>
</div>
<div xml:id="echoid-div564" type="section" level="1" n="90">
<head xml:id="echoid-head93" style="it" xml:space="preserve">Series octava.</head>
<p>
  <s xml:id="echoid-s15709" xml:space="preserve">{_cc_/_a_} + _b_ - _a_ = _n_.</s>
  <s xml:id="echoid-s15710" xml:space="preserve"/>
</p>
<note position="left" xml:space="preserve">Fig. 215.</note>
<p>
  <s xml:id="echoid-s15711" xml:space="preserve">_cc_ + _ba_ - _aa_ = _nn._</s>
  <s xml:id="echoid-s15712" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15713" xml:space="preserve">_cca_ + _baa_ - _a_<emph style="sub">3</emph> = _n_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15714" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15715" xml:space="preserve">_ccaa_ + _ba_<emph style="sub">3</emph> - _a_<emph style="sub">4</emph> = _n_<emph style="sub">4</emph>, &amp;</s>
  <s xml:id="echoid-s15716" xml:space="preserve">c.</s>
  <s xml:id="echoid-s15717" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div565" type="section" level="1" n="91">
<head xml:id="echoid-head94" style="it" xml:space="preserve">Series nona.</head>
<p>
  <s xml:id="echoid-s15718" xml:space="preserve">_a_ - _b_ - {_cc_/_a_} = _n._</s>
  <s xml:id="echoid-s15719" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15720" xml:space="preserve">_aa_ - _ba_ - _cc_ = _nn._</s>
  <s xml:id="echoid-s15721" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15722" xml:space="preserve">_a_<emph style="sub">3</emph> - _baa_ - _cca_ = _n_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15723" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15724" xml:space="preserve">_a_<emph style="sub">4</emph> - _ba_<emph style="sub">3</emph> - _ccaa_ = _n_<emph style="sub">4</emph>. </s>
  <s xml:id="echoid-s15725" xml:space="preserve">&amp;</s>
  <s xml:id="echoid-s15726" xml:space="preserve">c.</s>
  <s xml:id="echoid-s15727" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15728" xml:space="preserve">In recta AI ſumatur AB = _b_; </s>
  <s xml:id="echoid-s15729" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15730" xml:space="preserve">in AD ad ipſam AI perpen-<lb/>
<anchor type="note" xlink:label="note-0318-02a" xlink:href="note-0318-02"/>
diculari ſit AC = _c_; </s>
  <s xml:id="echoid-s15731" xml:space="preserve">fiant autem anguli IAR, ABS ſemirecti; <lb/></s>
  <s xml:id="echoid-s15732" xml:space="preserve">ducatúrque recta ZGK ad AI utcunque perpendicularis, ipſam BS <lb/>ſecans ad ξ; </s>
  <s xml:id="echoid-s15733" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15734" xml:space="preserve">ſit AG. </s>
  <s xml:id="echoid-s15735" xml:space="preserve">AC:</s>
  <s xml:id="echoid-s15736" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s15737" xml:space="preserve">ξ K; </s>
  <s xml:id="echoid-s15738" xml:space="preserve">tum per K intra angu-<lb/>lum DSB deſcribatur _byperbola_ KYHK; </s>
  <s xml:id="echoid-s15739" xml:space="preserve">ſint denuò curvæ CLHLλ, <lb/>AMHMμ, ANHNν tales, ut inter AG, GK ſint _media_ GL, _bime-_-<lb/>_dia_ GM, _trimedia_ GN; </s>
  <s xml:id="echoid-s15740" xml:space="preserve">hæ curvæ propoſito ſatisfacient; </s>
  <s xml:id="echoid-s15741" xml:space="preserve">conſtat <lb/>autem hoc ut in præcedente.</s>
  <s xml:id="echoid-s15742" xml:space="preserve"/>
</p>
<div xml:id="echoid-div565" type="float" level="2" n="1">
<note position="left" xlink:label="note-0318-02" xlink:href="note-0318-02a" xml:space="preserve">Fig. 215.</note>
</div>
</div>
<div xml:id="echoid-div567" type="section" level="1" n="92">
<head xml:id="echoid-head95" style="it" xml:space="preserve">Not.</head>
<p>
  <s xml:id="echoid-s15743" xml:space="preserve">1. </s>
  <s xml:id="echoid-s15744" xml:space="preserve">Curvæ CLH, AMH, ANH ad octavam ſeriem pertinent, re-<lb/>liquæ verò HLλ, HMμ, HN@, ad nonam.</s>
  <s xml:id="echoid-s15745" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15746" xml:space="preserve">2. </s>
  <s xml:id="echoid-s15747" xml:space="preserve">Quoad octavam ſeriem, ſi biſecetur AB in O, &amp; </s>
  <s xml:id="echoid-s15748" xml:space="preserve">ordinetur OT <lb/>ad curvam CLH eſt OT maxima; </s>
  <s xml:id="echoid-s15749" xml:space="preserve">ſin ſiat AP = {_b_/3} + √{_bb_/9} +
<pb o="141" file="0319" n="334" rhead=""/>
{_cc_/3}, ac ordinetur PV ad curvam AMH, erit PV maxima; </s>
  <s xml:id="echoid-s15750" xml:space="preserve">item ſi <lb/>AQ = {3/8}_b_ + √{9/64}_bb_ + {_cc_/2}, &amp; </s>
  <s xml:id="echoid-s15751" xml:space="preserve">ordinetur QX ad curvam ANH <lb/>erit QX maxima.</s>
  <s xml:id="echoid-s15752" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15753" xml:space="preserve">3. </s>
  <s xml:id="echoid-s15754" xml:space="preserve">Hinc, ſi in ſecundo harum gradu ſit _n_&amp;</s>
  <s xml:id="echoid-s15755" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s15756" xml:space="preserve">√ _cc_ + {_bb_/4}; </s>
  <s xml:id="echoid-s15757" xml:space="preserve">in ter-<lb/>tio ſi (poſito fore f = {_b_/3} + √{_bb_/9} + {_cc_/3}) ſit _n_<emph style="sub">3</emph> &amp;</s>
  <s xml:id="echoid-s15758" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s15759" xml:space="preserve">_ccf_ + _bff_ <lb/>- _f_<emph style="sub">3</emph>; </s>
  <s xml:id="echoid-s15760" xml:space="preserve">in quarto, ſi (poſito fore _g_ = {3/8}_b_ + √{9/64}_bb_ + {_cc_/2}) ſit _n_<emph style="sub">4</emph> <lb/>&amp;</s>
  <s xml:id="echoid-s15761" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s15762" xml:space="preserve">_ccgg_ + _bg_<emph style="sub">3</emph> - _g_<emph style="sub">4</emph>; </s>
  <s xml:id="echoid-s15763" xml:space="preserve">nulla datur radix; </s>
  <s xml:id="echoid-s15764" xml:space="preserve">nam his ſupp ſitis, <lb/>recta EF curvis non occurret, reſpectivè.</s>
  <s xml:id="echoid-s15765" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15766" xml:space="preserve">4. </s>
  <s xml:id="echoid-s15767" xml:space="preserve">Si fuerit Aφ = {_b_/4} + √{_bb_/16} + {_cc_/2}, &amp; </s>
  <s xml:id="echoid-s15768" xml:space="preserve">ordinetur φ Y; </s>
  <s xml:id="echoid-s15769" xml:space="preserve">erit Y <lb/>_Nodus_ curvarum; </s>
  <s xml:id="echoid-s15770" xml:space="preserve">unde ſi _n_ = Aφ; </s>
  <s xml:id="echoid-s15771" xml:space="preserve">erit Aφ una radicum in omni-<lb/>bus.</s>
  <s xml:id="echoid-s15772" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15773" xml:space="preserve">5. </s>
  <s xml:id="echoid-s15774" xml:space="preserve">Curva CLH, eſt _circumferentia Circuli_, cujus _Centrum_ O; <lb/></s>
  <s xml:id="echoid-s15775" xml:space="preserve">reliquæ AMH, ANH ſunt _Cycliformes_.</s>
  <s xml:id="echoid-s15776" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15777" xml:space="preserve">6. </s>
  <s xml:id="echoid-s15778" xml:space="preserve">Peculiare eſt in ſecundo gradu, quòd ſi n&amp;</s>
  <s xml:id="echoid-s15779" xml:space="preserve">lt;</s>
  <s xml:id="echoid-s15780" xml:space="preserve">c, detur una tan-<lb/>tùm radix.</s>
  <s xml:id="echoid-s15781" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15782" xml:space="preserve">7. </s>
  <s xml:id="echoid-s15783" xml:space="preserve">In hac radicum maxima (quæ &amp; </s>
  <s xml:id="echoid-s15784" xml:space="preserve">minima eſt in nona ſerie) eſt <lb/>AH = {_b_/2} + √{_bb_/4} + _cc_.</s>
  <s xml:id="echoid-s15785" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15786" xml:space="preserve">8. </s>
  <s xml:id="echoid-s15787" xml:space="preserve">Curva HL λ eſt _hyperbola æquilatera_, cujus _ſemiaxis_ OH; </s>
  <s xml:id="echoid-s15788" xml:space="preserve">re-<lb/>liquæ HMμ, HNν ſunt _hyperboliformes_; </s>
  <s xml:id="echoid-s15789" xml:space="preserve">unde patet in ſerie nona <lb/>ſemper unam, &amp; </s>
  <s xml:id="echoid-s15790" xml:space="preserve">hanc unicam radicem haberi.</s>
  <s xml:id="echoid-s15791" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div568" type="section" level="1" n="93">
<head xml:id="echoid-head96" style="it" xml:space="preserve">Series decima.</head>
<note position="right" xml:space="preserve">Fig. 216.</note>
<p>
  <s xml:id="echoid-s15792" xml:space="preserve">_a_ + _b_ - {_cc_/_a_} = _n_.</s>
  <s xml:id="echoid-s15793" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15794" xml:space="preserve">_aa_ + _ba_ - _cc_ = _nn_.</s>
  <s xml:id="echoid-s15795" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15796" xml:space="preserve">_a_<emph style="sub">3</emph> + _baa_ - _cca_ = _n_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15797" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15798" xml:space="preserve">_a_<emph style="sub">4</emph> + _ba_<emph style="sub">3</emph>-_ccaa_ = _n_<emph style="sub">4</emph>, &amp;</s>
  <s xml:id="echoid-s15799" xml:space="preserve">c.</s>
  <s xml:id="echoid-s15800" xml:space="preserve"/>
</p>
<pb o="142" file="0320" n="335" rhead=""/>
</div>
<div xml:id="echoid-div569" type="section" level="1" n="94">
<head xml:id="echoid-head97" style="it" xml:space="preserve">Series undecima.</head>
<p>
  <s xml:id="echoid-s15801" xml:space="preserve">{_cc_/_a_} - _b_ - _a_ = _n_.</s>
  <s xml:id="echoid-s15802" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15803" xml:space="preserve">_cc_ - _ba_ - _aa_ = _nn_.</s>
  <s xml:id="echoid-s15804" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15805" xml:space="preserve">_cca_ - _baa_ - _a_<emph style="sub">3</emph> = _n_<emph style="sub">3</emph>.</s>
  <s xml:id="echoid-s15806" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15807" xml:space="preserve">_ccaa_ - _ba_<emph style="sub">3</emph> - _a_<emph style="sub">4</emph> = _n_<emph style="sub">4</emph>, &amp;</s>
  <s xml:id="echoid-s15808" xml:space="preserve">c.</s>
  <s xml:id="echoid-s15809" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15810" xml:space="preserve">In recta BAH ſumatur BA = _b_; </s>
  <s xml:id="echoid-s15811" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15812" xml:space="preserve">in AD ad AH perpendi-<lb/>
<anchor type="note" xlink:label="note-0320-01a" xlink:href="note-0320-01"/>
culari ſit AC = _c_; </s>
  <s xml:id="echoid-s15813" xml:space="preserve">ſintque anguli HAR, HBS ſemirecti; </s>
  <s xml:id="echoid-s15814" xml:space="preserve">tum <lb/>utcunque ductâ GK ξ ad AH perpendiculari (quæ ipſam BS <lb/>ſecet in ξ; </s>
  <s xml:id="echoid-s15815" xml:space="preserve">ſit AG. </s>
  <s xml:id="echoid-s15816" xml:space="preserve">AC:</s>
  <s xml:id="echoid-s15817" xml:space="preserve">: AC. </s>
  <s xml:id="echoid-s15818" xml:space="preserve">ξK; </s>
  <s xml:id="echoid-s15819" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15820" xml:space="preserve">per K intra _aſymptotos_ <lb/>VD, VS deſcribatur _hyperbola_ KYHK; </s>
  <s xml:id="echoid-s15821" xml:space="preserve">ſint demum curvæ CLHLλ, <lb/>AMHMμ, ANHNν tales, ut inter AG (vel GZ) &amp; </s>
  <s xml:id="echoid-s15822" xml:space="preserve">GK ſint _me_-<lb/>_dia_ GL, _bimedia_ GM, _trimedia_ GN; </s>
  <s xml:id="echoid-s15823" xml:space="preserve">hæ propoſito ſervient. </s>
  <s xml:id="echoid-s15824" xml:space="preserve">id <lb/>quod conſtat, ut in præcedentibus.</s>
  <s xml:id="echoid-s15825" xml:space="preserve"/>
</p>
<div xml:id="echoid-div569" type="float" level="2" n="1">
<note position="left" xlink:label="note-0320-01" xlink:href="note-0320-01a" xml:space="preserve">Fig. 216.</note>
</div>
</div>
<div xml:id="echoid-div571" type="section" level="1" n="95">
<head xml:id="echoid-head98" style="it" xml:space="preserve">Not.</head>
<p>
  <s xml:id="echoid-s15826" xml:space="preserve">1. </s>
  <s xml:id="echoid-s15827" xml:space="preserve">Curvæ HLλ, HMμ, HNν ad decimam ſeriem pertinent; <lb/></s>
  <s xml:id="echoid-s15828" xml:space="preserve">reliquæ CLH, AMH, ANH ad undecimam.</s>
  <s xml:id="echoid-s15829" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15830" xml:space="preserve">2. </s>
  <s xml:id="echoid-s15831" xml:space="preserve">Curva HL λ eſt _hyperbola æquilatera_, &amp; </s>
  <s xml:id="echoid-s15832" xml:space="preserve">curva CLH _circula_-<lb/>_ris circumferentiæ_ pars; </s>
  <s xml:id="echoid-s15833" xml:space="preserve">utriuſque commune centrum eſt O, ipſam AB <lb/>biſecans (unde AH = √{_bb_/4} + _cc_: </s>
  <s xml:id="echoid-s15834" xml:space="preserve">- {_b_/2})</s>
</p>
<p>
  <s xml:id="echoid-s15835" xml:space="preserve">3. </s>
  <s xml:id="echoid-s15836" xml:space="preserve">In decima ſerie radix una ſemper habetur, &amp; </s>
  <s xml:id="echoid-s15837" xml:space="preserve">unica; </s>
  <s xml:id="echoid-s15838" xml:space="preserve">in undeci-<lb/>ma nunc duæ, nunc una, ſubinde nulla.</s>
  <s xml:id="echoid-s15839" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15840" xml:space="preserve">4. </s>
  <s xml:id="echoid-s15841" xml:space="preserve">Aφ = {_cc_/_b_}; </s>
  <s xml:id="echoid-s15842" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15843" xml:space="preserve">Aψ = √{_bb_/16} + {_cc_/2}: </s>
  <s xml:id="echoid-s15844" xml:space="preserve">- {_b_/4}; </s>
  <s xml:id="echoid-s15845" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15846" xml:space="preserve">ordinentur <lb/>φ Y, ψ X; </s>
  <s xml:id="echoid-s15847" xml:space="preserve">puncta Y, X ſunt nodi curvarum.</s>
  <s xml:id="echoid-s15848" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15849" xml:space="preserve">5. </s>
  <s xml:id="echoid-s15850" xml:space="preserve">In undecimæ ſecundo gradu ordinata AC eſt maxìma; </s>
  <s xml:id="echoid-s15851" xml:space="preserve">ſin AP <lb/>= √{_bb_/9} + {_cc_/3}: </s>
  <s xml:id="echoid-s15852" xml:space="preserve">- {_b_/3}; </s>
  <s xml:id="echoid-s15853" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15854" xml:space="preserve">à P ad curvam AMH ordinetur Pγ, <lb/>hæc maxima erit; </s>
  <s xml:id="echoid-s15855" xml:space="preserve">item ſi AQ = √{9_bb_/64} + {_cc_/2}: </s>
  <s xml:id="echoid-s15856" xml:space="preserve">- {3_b_/8}; </s>
  <s xml:id="echoid-s15857" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15858" xml:space="preserve">à
<pb o="143" file="0321" n="336" rhead=""/>
Qad curvam AN H ordinetur Q δ, hæc etiam maxima erit; </s>
  <s xml:id="echoid-s15859" xml:space="preserve">unde <lb/>de radicum limitibus fiet judicium; </s>
  <s xml:id="echoid-s15860" xml:space="preserve">ut in iis, quæ ad ſeriem octavam <lb/>ſunt adnotata.</s>
  <s xml:id="echoid-s15861" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div572" type="section" level="1" n="96">
<head xml:id="echoid-head99" style="it" xml:space="preserve">Series duodecima</head>
<note position="right" xml:space="preserve">Fig. 217</note>
<p>
  <s xml:id="echoid-s15862" xml:space="preserve">_a_ - _b_ + {_cc_/_a_} = _n_.</s>
  <s xml:id="echoid-s15863" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15864" xml:space="preserve">_aa_ - _ba_ + _cc_ = _nn_.</s>
  <s xml:id="echoid-s15865" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15866" xml:space="preserve">_a<emph style="sub">3</emph>_ - _baa_ + _cca_ = _n<emph style="sub">3</emph>_.</s>
  <s xml:id="echoid-s15867" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15868" xml:space="preserve">_a<emph style="sub">4</emph>_ - _ba<emph style="sub">3</emph>_ + _ccaa_ = _n<emph style="sub">4</emph>_, &amp;</s>
  <s xml:id="echoid-s15869" xml:space="preserve">c.</s>
  <s xml:id="echoid-s15870" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div573" type="section" level="1" n="97">
<head xml:id="echoid-head100" style="it" xml:space="preserve">Series decima tertia</head>
<p>
  <s xml:id="echoid-s15871" xml:space="preserve">_b_ - _a_ - {_cc_/_a_} = _n_.</s>
  <s xml:id="echoid-s15872" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15873" xml:space="preserve">_ba_ - _aa_ - _cc_ = _nn_.</s>
  <s xml:id="echoid-s15874" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15875" xml:space="preserve">_baa_ - _a<emph style="sub">3</emph>_ - _cca_ = _n<emph style="sub">3</emph>_.</s>
  <s xml:id="echoid-s15876" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15877" xml:space="preserve">_ba<emph style="sub">3</emph>_ - _a<emph style="sub">4</emph>_ - _ccaa_=_n<emph style="sub">4</emph>_, &amp;</s>
  <s xml:id="echoid-s15878" xml:space="preserve">c.</s>
  <s xml:id="echoid-s15879" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15880" xml:space="preserve">Pro his, Sit AB=_b_; </s>
  <s xml:id="echoid-s15881" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15882" xml:space="preserve">AC = _c_; </s>
  <s xml:id="echoid-s15883" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15884" xml:space="preserve">angulus. </s>
  <s xml:id="echoid-s15885" xml:space="preserve">AB S ſemire-<lb/>
<anchor type="note" xlink:label="note-0321-02a" xlink:href="note-0321-02"/>
ctus, &amp; </s>
  <s xml:id="echoid-s15886" xml:space="preserve">G ξ ad AB utcunque perpendicularis, &amp; </s>
  <s xml:id="echoid-s15887" xml:space="preserve">AG . </s>
  <s xml:id="echoid-s15888" xml:space="preserve">AC :</s>
  <s xml:id="echoid-s15889" xml:space="preserve">: AC. <lb/></s>
  <s xml:id="echoid-s15890" xml:space="preserve">ξ K; </s>
  <s xml:id="echoid-s15891" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15892" xml:space="preserve">KH KI K _byperbola aſymptotis_ SA , SB deſcripta; </s>
  <s xml:id="echoid-s15893" xml:space="preserve">denuò <lb/>curvæ CLHLILλ, AMHMIMμ, ANHNINν tales ſint, ut inter AG, <lb/>GK ſit _media_ GL , _bimedia_ GM , _trimedia_ GN .</s>
  <s xml:id="echoid-s15894" xml:space="preserve"/>
</p>
<div xml:id="echoid-div573" type="float" level="2" n="1">
<note position="right" xlink:label="note-0321-02" xlink:href="note-0321-02a" xml:space="preserve">Fig. 217.</note>
</div>
</div>
<div xml:id="echoid-div575" type="section" level="1" n="98">
<head xml:id="echoid-head101" style="it" xml:space="preserve">Not.</head>
<p>
  <s xml:id="echoid-s15895" xml:space="preserve">1. </s>
  <s xml:id="echoid-s15896" xml:space="preserve">Curvæ CLH, AMH, ANH, atque curvæ IL λ, IM μ, <lb/>IN ν ad ſeriem duodecimam ſpectant, verùm intermediæ curvæ HLI, <lb/>HMI, HNI ad decimam tertiam.</s>
  <s xml:id="echoid-s15897" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15898" xml:space="preserve">2. </s>
  <s xml:id="echoid-s15899" xml:space="preserve">Curvæ CLH, IL λ ſunt _hyperbolæ æquilateræ_, quarum com-<lb/>mune _centrum_ O (rectam AB biſecans) &amp; </s>
  <s xml:id="echoid-s15900" xml:space="preserve">_ſemiaxis_ OH (vel OI) <lb/>= √ AO q. </s>
  <s xml:id="echoid-s15901" xml:space="preserve">- AC q reliquæ tales ſunt, quales figura monſtrat.</s>
  <s xml:id="echoid-s15902" xml:space="preserve"/>
</p>
<pb o="144" file="0322" n="337" rhead=""/>
<p>
  <s xml:id="echoid-s15903" xml:space="preserve">3. </s>
  <s xml:id="echoid-s15904" xml:space="preserve">Curva HLLI eſt _ſemicirculus_; </s>
  <s xml:id="echoid-s15905" xml:space="preserve">reliquas itidem oſtentat <lb/>Schema.</s>
  <s xml:id="echoid-s15906" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15907" xml:space="preserve">4. </s>
  <s xml:id="echoid-s15908" xml:space="preserve">Si A ζ = {_cc_/_b_}; </s>
  <s xml:id="echoid-s15909" xml:space="preserve">A Ψ = {_b_/4} - √ {_bb_/16} - {_cc_/2}; </s>
  <s xml:id="echoid-s15910" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15911" xml:space="preserve">A φ = {_b_/4} + <lb/>√ {_bb_/16} - {_cc_/2}; </s>
  <s xml:id="echoid-s15912" xml:space="preserve">ordinentúrque rectæ ζ V, ψ X, φ Y; </s>
  <s xml:id="echoid-s15913" xml:space="preserve">erunt puncta V, <lb/>X, Y _nodi_ curvarum (ſi _b_ &amp;</s>
  <s xml:id="echoid-s15914" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s15915" xml:space="preserve">√ 8 _c c_, deerunt _nodi_ X, Y; </s>
  <s xml:id="echoid-s15916" xml:space="preserve">ſi _b_ = √ <lb/>8 _c c_; </s>
  <s xml:id="echoid-s15917" xml:space="preserve">ii coaleſcent).</s>
  <s xml:id="echoid-s15918" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15919" xml:space="preserve">5. </s>
  <s xml:id="echoid-s15920" xml:space="preserve">Ordinatarum ad curvam CL H _maxima_ eſt ipſa AC ; </s>
  <s xml:id="echoid-s15921" xml:space="preserve">ſin AP <lb/>= {_b_/3} - √ {_bb_/9} - {_cc_/3}, &amp; </s>
  <s xml:id="echoid-s15922" xml:space="preserve">ordinetur P γ ad curvam AM H; </s>
  <s xml:id="echoid-s15923" xml:space="preserve">erit <lb/>P γ _maxima_; </s>
  <s xml:id="echoid-s15924" xml:space="preserve">item ſi AQ = {3/8} _b_ - √ {@9/64} _b b_ - {_cc_/2}; </s>
  <s xml:id="echoid-s15925" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15926" xml:space="preserve">ordinetur <lb/>Q δ ad curvam AN H, erit Q δ _maxima_.</s>
  <s xml:id="echoid-s15927" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15928" xml:space="preserve">6. </s>
  <s xml:id="echoid-s15929" xml:space="preserve">Ordinatarum ad curvam HLLI _maxima_ eſt ipſa OT ; </s>
  <s xml:id="echoid-s15930" xml:space="preserve">ſin AP <lb/>= {_b_/3} + √ {_bb_/9} - {_cc_/3}, &amp; </s>
  <s xml:id="echoid-s15931" xml:space="preserve">ad curvam HM I ordinetur _p g_, erit _p g_ <lb/>_maxima_; </s>
  <s xml:id="echoid-s15932" xml:space="preserve">item ſi A q = {3/8} _b_ + √ {9/64} _b b_ - {_cc_/2}; </s>
  <s xml:id="echoid-s15933" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15934" xml:space="preserve">ordinetur _q d_ <lb/>ad curvam HN I, erit _q d maxima_.</s>
  <s xml:id="echoid-s15935" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15936" xml:space="preserve">7. </s>
  <s xml:id="echoid-s15937" xml:space="preserve">Hinc radicum limites dignoſcentur, ut innuitur in iis, quæ ad <lb/>octavam ſeriem animadverſa ſunt.</s>
  <s xml:id="echoid-s15938" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15939" xml:space="preserve">8. </s>
  <s xml:id="echoid-s15940" xml:space="preserve">Patet in Serie duodecima nunc tres, modo duas, ſemper unam <lb/>radicem haberi; </s>
  <s xml:id="echoid-s15941" xml:space="preserve">in decima tertia verò ſubinde duas, aliquando tantùm <lb/>unam, interdum nullam haberi.</s>
  <s xml:id="echoid-s15942" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15943" xml:space="preserve">9. </s>
  <s xml:id="echoid-s15944" xml:space="preserve">Et hæc quidem conſtant poſito fore {_b_/2}&amp;</s>
  <s xml:id="echoid-s15945" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s15946" xml:space="preserve">_c_; </s>
  <s xml:id="echoid-s15947" xml:space="preserve">at ſi {_b_/2} = β; <lb/></s>
  <s xml:id="echoid-s15948" xml:space="preserve">evaneſcet Series decima tertia; </s>
  <s xml:id="echoid-s15949" xml:space="preserve">coaleſcent puncta H, O, I; </s>
  <s xml:id="echoid-s15950" xml:space="preserve">recta AB <lb/>_byperbolam_ KK K tanget; </s>
  <s xml:id="echoid-s15951" xml:space="preserve">curvæque CL H, IL λ in rectas lineas <lb/>degenerabunt.</s>
  <s xml:id="echoid-s15952" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15953" xml:space="preserve">10. </s>
  <s xml:id="echoid-s15954" xml:space="preserve">Sin {_b_/2} &amp;</s>
  <s xml:id="echoid-s15955" xml:space="preserve">lt; </s>
  <s xml:id="echoid-s15956" xml:space="preserve">_c_; </s>
  <s xml:id="echoid-s15957" xml:space="preserve">etiam evaneſcit Series decima tertia; </s>
  <s xml:id="echoid-s15958" xml:space="preserve">_byperbola_ KKK <lb/>tota infra rectam AB jacente; </s>
  <s xml:id="echoid-s15959" xml:space="preserve">quo caſu curva CL L erit hyperbola <lb/>æquilatera, habens centrum O, ſemiaxem (ipſi AB perpendicula-<lb/>rem) OT = √ AC q - AO q; </s>
  <s xml:id="echoid-s15960" xml:space="preserve">tunc &amp; </s>
  <s xml:id="echoid-s15961" xml:space="preserve">curvæ AM M, AN N <lb/>
<anchor type="note" xlink:label="note-0322-01a" xlink:href="note-0322-01"/>
ad infinitum procurrent, ſic ut æquationes, quæ in Serie duodecima, <lb/>unam ſemper, &amp; </s>
  <s xml:id="echoid-s15962" xml:space="preserve">unicam radicem obtineant. </s>
  <s xml:id="echoid-s15963" xml:space="preserve">Hæc ſuffecerit inſinu-<lb/>âſſe; </s>
  <s xml:id="echoid-s15964" xml:space="preserve">quin &amp; </s>
  <s xml:id="echoid-s15965" xml:space="preserve">rem totam hactenus particulatim attigiſſe. </s>
  <s xml:id="echoid-s15966" xml:space="preserve">Subnecte-<lb/>mus autem notas quaſdam magìs generales.</s>
  <s xml:id="echoid-s15967" xml:space="preserve"/>
</p>
<div xml:id="echoid-div575" type="float" level="2" n="1">
<note position="left" xlink:label="note-0322-01" xlink:href="note-0322-01a" xml:space="preserve">Fig. 218.</note>
</div>
<pb o="145" file="0323" n="338" rhead=""/>
<p>
  <s xml:id="echoid-s15968" xml:space="preserve">In _pramiſſas explicationes_ animadvertatur generatim.</s>
  <s xml:id="echoid-s15969" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s15970" xml:space="preserve">1. </s>
  <s xml:id="echoid-s15971" xml:space="preserve">Propoſitam quamvis æquationem explicans _@μγνα_ deſignatur <lb/>hoc modo: </s>
  <s xml:id="echoid-s15972" xml:space="preserve">proponatur, exempli causâ, _æquatio a<emph style="sub">5</emph>_ + _ba<emph style="sub">4</emph>_ + _cca<emph style="sub">3</emph>_ <lb/>- _d<emph style="sub">3</emph>aa_ - _f<emph style="sub">4</emph>a_ = _n<emph style="sub">5</emph>_; </s>
  <s xml:id="echoid-s15973" xml:space="preserve">In recta indefinitè protenſa HI deſignetur pun-<lb/>
<anchor type="note" xlink:label="note-0323-01a" xlink:href="note-0323-01"/>
ctum A, pro radicum termino, vel origine; </s>
  <s xml:id="echoid-s15974" xml:space="preserve">tum arbitrariè ſumptâ <lb/>AG pro indeterminatâ radice _a_; </s>
  <s xml:id="echoid-s15975" xml:space="preserve">fiat GK æqualis primo feriei pro-<lb/>poſitam æquationem continentis gradu; </s>
  <s xml:id="echoid-s15976" xml:space="preserve">nempe ſit hîc GK = _a_ + _b_ <lb/>+ {_cc_/_a_} - {_d<emph style="sub">3</emph>_/_aa_} - {_f<emph style="sub">+</emph>_/_a<emph style="sub">3</emph>_} (utique rationem _a_ ad _c_ ſemel continuando fit <lb/>{_cc_/_a_}; </s>
  <s xml:id="echoid-s15977" xml:space="preserve">rationem _a_ ad _d_ bis continuando fit {_d<emph style="sub">3</emph>_/_aa_}; </s>
  <s xml:id="echoid-s15978" xml:space="preserve">acità porrò) tum inter <lb/>AG, GK tot mediarum proportionalium, quot æquationis propoſitæ <lb/>gradus exigit (is autem à pura quæſitæ radicis poteſtate indicatur) in <lb/>hoc nempe caſu quatuor mediarum proportionalium prima ſit GO; <lb/></s>
  <s xml:id="echoid-s15979" xml:space="preserve">per ejuſmodi puncta O traducta curva AOO propoſito deſerviet.</s>
  <s xml:id="echoid-s15980" xml:space="preserve"/>
</p>
<div xml:id="echoid-div576" type="float" level="2" n="2">
<note position="right" xlink:label="note-0323-01" xlink:href="note-0323-01a" xml:space="preserve">Fig. 219.</note>
</div>
<p>
  <s xml:id="echoid-s15981" xml:space="preserve">2. </s>
  <s xml:id="echoid-s15982" xml:space="preserve">De radicibus falſis, ſeu negativis nihil attigimus ſuprà; </s>
  <s xml:id="echoid-s15983" xml:space="preserve">cæte-<lb/>rùm eæ reperiuntur hoc modo. </s>
  <s xml:id="echoid-s15984" xml:space="preserve">Æquationi propoſitæ ſubrogetur <lb/>altera, cujus in locis paribus (etiam vacuos locos adnumerando) <lb/>ſigna ſunt illis contraria, quæ habet æquatio propoſita; </s>
  <s xml:id="echoid-s15985" xml:space="preserve">erunt hu-<lb/>juſce _ſubdititiæ æquationis_ radices veræ, ſeu poſitivæ ipſius propoſitæ <lb/>æquationis radices falſæ, ſeu negativæ. </s>
  <s xml:id="echoid-s15986" xml:space="preserve">_Exemplo_ ſit _æquatio a<emph style="sub">3</emph>_ + _baa_ <lb/>= _n<emph style="sub">3</emph>_; </s>
  <s xml:id="echoid-s15987" xml:space="preserve">vel _a<emph style="sub">3</emph>_ + _baa*_ - _n<emph style="sub">3</emph>_ = _o_. </s>
  <s xml:id="echoid-s15988" xml:space="preserve">Subrogetur _a<emph style="sub">3</emph>_ - _baa<emph style="sub">*</emph>_ + _n<emph style="sub">3</emph>_ = _o_; </s>
  <s xml:id="echoid-s15989" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s15990" xml:space="preserve"><lb/>hujus, + utì ſuprà edoctum, veræ radices deſignentur, hæ _propoſitæ_ <lb/>
<anchor type="note" xlink:label="note-0323-02a" xlink:href="note-0323-02"/>
_aquationis_ falſæ erunt. </s>
  <s xml:id="echoid-s15991" xml:space="preserve">Rurſus ſit _a<emph style="sub">3</emph>_ - _baa_ = _n<emph style="sub">3</emph>_; </s>
  <s xml:id="echoid-s15992" xml:space="preserve">vel _a<emph style="sub">3</emph>_ - _baa_ - _n<emph style="sub">3</emph>_ <lb/>= _o_; </s>
  <s xml:id="echoid-s15993" xml:space="preserve">ſubſtituatur æquatio _a<emph style="sub">3</emph>_ + _baa_ + _n<emph style="sub">3</emph>_ = _o_; </s>
  <s xml:id="echoid-s15994" xml:space="preserve">hæc nullam veram <lb/>radicem obtinet; </s>
  <s xml:id="echoid-s15995" xml:space="preserve">ergò nec _æquatio propoſita_ falſam admittit.</s>
  <s xml:id="echoid-s15996" xml:space="preserve"/>
</p>
<div xml:id="echoid-div577" type="float" level="2" n="3">
<note position="right" xlink:label="note-0323-02" xlink:href="note-0323-02a" xml:space="preserve">_+ In Serie 3_.</note>
</div>
<p>
  <s xml:id="echoid-s15997" xml:space="preserve">3. </s>
  <s xml:id="echoid-s15998" xml:space="preserve">Quinimò datâ verâ radice quâpiam, depreſſioris gradûs æqua-<lb/>tio quædam ſalſis reperiendis inſerviet, qualis ità determinatur. </s>
  <s xml:id="echoid-s15999" xml:space="preserve">Pro-<lb/>ponatur æquatio quævis, puta _a<emph style="sub">3</emph>_ + _baa_ = _n<emph style="sub">3</emph>_; </s>
  <s xml:id="echoid-s16000" xml:space="preserve">cujus nota ſit radix una, <lb/>quæ vocetur _f_. </s>
  <s xml:id="echoid-s16001" xml:space="preserve">Conſtruatur æquatio planè ſimilis propoſitæ, eáſ-<lb/>demque _coefficientes_ habens, tantum pro _a_ ſubſtituendo _f_; </s>
  <s xml:id="echoid-s16002" xml:space="preserve">nempe <lb/>_f<emph style="sub">3</emph>_ + _bff_ = _n<emph style="sub">3</emph>_. </s>
  <s xml:id="echoid-s16003" xml:space="preserve">ergo _a<emph style="sub">3</emph>_ + _baa_ = _n<emph style="sub">3</emph>_ = _f<emph style="sub">3</emph>_ + _bff_; </s>
  <s xml:id="echoid-s16004" xml:space="preserve">adeóque <lb/>_a<emph style="sub">3</emph>_ + _baa_ - _f<emph style="sub">3</emph>_ - _bff_ = _o_. </s>
  <s xml:id="echoid-s16005" xml:space="preserve">dividatur hæc æquatio (id quod ſem-<lb/>per fieri poteſt) per _a_ - _f_; </s>
  <s xml:id="echoid-s16006" xml:space="preserve">proveniet _a a_ { + _ba_ + _bf_ + _fa_ + _ff_} = _o_; </s>
  <s xml:id="echoid-s16007" xml:space="preserve">cujus æ-<lb/>quationes eædem erunt cum reliquis æquationis propoſitæ radicibus; <lb/></s>
  <s xml:id="echoid-s16008" xml:space="preserve">quæ proinde duas colligitur radices falſas habere; </s>
  <s xml:id="echoid-s16009" xml:space="preserve">itaque mutatis loco-<lb/>rum parìum ſignis, ut ità fiat _a a_ { - _ba_ + _bf_/ - _fa_ + _ff_} = _o_; </s>
  <s xml:id="echoid-s16010" xml:space="preserve">hujus æquationis
<pb o="146" file="0324" n="339" rhead=""/>
veræ radices propoſitæ falſas exhibent. </s>
  <s xml:id="echoid-s16011" xml:space="preserve">Hic inſuper modus æquatio-<lb/>nis propoſitæ, quatenus illa ex aliarum in ſe ductu provenit, conſtitutio-<lb/>nem oſtendit.</s>
  <s xml:id="echoid-s16012" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16013" xml:space="preserve">4. </s>
  <s xml:id="echoid-s16014" xml:space="preserve">Radices maximæ &amp; </s>
  <s xml:id="echoid-s16015" xml:space="preserve">minimæ deprehenduntur in quacunque ſe-<lb/>rie ponendo (quovis in gradu ſeriei) fore n = o; </s>
  <s xml:id="echoid-s16016" xml:space="preserve">ut in octava ſerie ſit <lb/>_ba_ - _aa_ + _cc_ = _o_; </s>
  <s xml:id="echoid-s16017" xml:space="preserve">adeóque _cc_ = _aa_ - _ba_, erit _a_ ( = {_b_/2} + √ {_bb_/4} + <lb/>_cc_) _maxima radix_; </s>
  <s xml:id="echoid-s16018" xml:space="preserve">item in Serie duodecima ſit _aa_ - _ba_ + _cc_ = _o_; <lb/></s>
  <s xml:id="echoid-s16019" xml:space="preserve">unde _cc_ = _ba_ - _aa_; </s>
  <s xml:id="echoid-s16020" xml:space="preserve">erit _a_ ( = {_b_/2} + √{_bb_/4} - _cc_) _radix maxima_; </s>
  <s xml:id="echoid-s16021" xml:space="preserve"><lb/>&amp; </s>
  <s xml:id="echoid-s16022" xml:space="preserve">_a_ ( = {_b_/2} - √ {_bb_/4} - _cc_) _radix minima_.</s>
  <s xml:id="echoid-s16023" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16024" xml:space="preserve">5. </s>
  <s xml:id="echoid-s16025" xml:space="preserve">_Curvaram nodi_, vel _interſectiones_ innoteſcunt, cujuſvis in Seriei <lb/>quovis gradu, ponendo fore _a_ = _n_; </s>
  <s xml:id="echoid-s16026" xml:space="preserve">ut in octava Serie, ubi _ba_ - _aa_ <lb/>+ _cc_ = _nn_, ſit _a_ = _n_; </s>
  <s xml:id="echoid-s16027" xml:space="preserve">ergò _ba_ - _aa_ + _cc_ = _aa_; </s>
  <s xml:id="echoid-s16028" xml:space="preserve">vel _cc_ = 2_aa_ - _ba_; <lb/></s>
  <s xml:id="echoid-s16029" xml:space="preserve">vel {_cc_/2} = _aa_ - {_ba_/2}; </s>
  <s xml:id="echoid-s16030" xml:space="preserve">quare _a_ = {_b_/4} + √ {_bb_/16} + {_cc_/2}. </s>
  <s xml:id="echoid-s16031" xml:space="preserve">Item in Se-<lb/>rie duodecima, ubi _aa_ - _ba_ + _cc_ = _nn_ = _aa_; </s>
  <s xml:id="echoid-s16032" xml:space="preserve">erit ideò _cc_ = _ba_; </s>
  <s xml:id="echoid-s16033" xml:space="preserve">acinde <lb/>_a_ = {_cc_/_b_}.</s>
  <s xml:id="echoid-s16034" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16035" xml:space="preserve">6. </s>
  <s xml:id="echoid-s16036" xml:space="preserve">_Ordinatæ maxima, mini@æque_ variis nodis, methodiſque paſ-<lb/>ſim notis inveſtigantur; </s>
  <s xml:id="echoid-s16037" xml:space="preserve">ego ſimul illas atque curvarum _tangentes_ <lb/>unà operâ ſic determino. </s>
  <s xml:id="echoid-s16038" xml:space="preserve">Sit curva A γ H, ad Seriem undecimam <lb/>pertinens, ejuſque gradum, cujus æquatio eſt _cca_ - _baa_ - _a<emph style="sub">3</emph>_ = _x<emph style="sub">3</emph>_; <lb/></s>
  <s xml:id="echoid-s16039" xml:space="preserve">
<anchor type="note" xlink:label="note-0324-01a" xlink:href="note-0324-01"/>
poſito γ T curvam tangere, &amp; </s>
  <s xml:id="echoid-s16040" xml:space="preserve">γ P ad AH ordinari, reperio (de ſu-<lb/>pra monſtratis) fore PT = {_3n<emph style="sub">3</emph>_/_3aa_ + _2ba_ - _cc_}, tum conſidero, ſi or-<lb/>dinata P γ ſit maxima, fore tangentem ipſi HA parallelam, ſeu rectam <lb/>PT eſſe infinitam; </s>
  <s xml:id="echoid-s16041" xml:space="preserve">quare cùm ſit _n<emph style="sub">3</emph>_ = PT x: </s>
  <s xml:id="echoid-s16042" xml:space="preserve">_3aa_ + _2ba_ - _cc_; </s>
  <s xml:id="echoid-s16043" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s16044" xml:space="preserve">_n_ <lb/>ſit finita, patet eſſe _3aa_ + _2ba_ - _cc_ = _o_; </s>
  <s xml:id="echoid-s16045" xml:space="preserve">vel _aa_ + {2/3}_ba_ = {_cc_/3}; </s>
  <s xml:id="echoid-s16046" xml:space="preserve">adeó-<lb/>que √: </s>
  <s xml:id="echoid-s16047" xml:space="preserve">{_bb_/9} + {_cc_/3}: </s>
  <s xml:id="echoid-s16048" xml:space="preserve">- {_b_/3} = _a_ = AP.</s>
  <s xml:id="echoid-s16049" xml:space="preserve"/>
</p>
<div xml:id="echoid-div578" type="float" level="2" n="4">
<note position="left" xlink:label="note-0324-01" xlink:href="note-0324-01a" xml:space="preserve">Fig. 220.</note>
</div>
<p>
  <s xml:id="echoid-s16050" xml:space="preserve">7. </s>
  <s xml:id="echoid-s16051" xml:space="preserve">Adnoto demùm è _maximis_ &amp; </s>
  <s xml:id="echoid-s16052" xml:space="preserve">_minimis ordinatis_ radicum li-<lb/>mites derivari; </s>
  <s xml:id="echoid-s16053" xml:space="preserve">nempe ſi reperiatur ad maximam ordinatam pertinen-<lb/>tis radicis (velut ipſius AP in exemplo proximè ſuperiori) valor, &amp; </s>
  <s xml:id="echoid-s16054" xml:space="preserve"><lb/>?</s>
  <s xml:id="echoid-s16055" xml:space="preserve">?is ubique in æ quatione pro ipsâ _a_ ſubſtituatur, ſi quod provenit, de-<lb/>ficiat ab _bomogeneo_ (quod vocant) _comparationis, problemn_ conſtrui
<pb o="147" file="0325" n="340" rhead=""/>
nequit, aut ſaltem radicibus aliquot caret, quas æquationis gradus &amp; </s>
  <s xml:id="echoid-s16056" xml:space="preserve"><lb/>ſpecies præ ſe ferunt. </s>
  <s xml:id="echoid-s16057" xml:space="preserve">Eadem _minimarum_ eſt ratio; </s>
  <s xml:id="echoid-s16058" xml:space="preserve">tantùm ibi <lb/>proveniens _ſumma_ debet _homogeneum_ illud excedere, quò radix ali-<lb/>qua, vel omnes habeantur. </s>
  <s xml:id="echoid-s16059" xml:space="preserve">_Exempla_ comparent in præmiſſis. </s>
  <s xml:id="echoid-s16060" xml:space="preserve">Hîc <lb/>itaque ſubſiſto.</s>
  <s xml:id="echoid-s16061" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div580" type="section" level="1" n="99">
<head xml:id="echoid-head102" style="it" xml:space="preserve">Laus DEOO ptimo Maximo.</head>
<head xml:id="echoid-head103" style="it" xml:space="preserve">FINIS.</head>
<pb file="0326" n="341"/>
</div>
<div xml:id="echoid-div581" type="section" level="1" n="100">
<head xml:id="echoid-head104" xml:space="preserve">ERRATA</head>
<p>
  <s xml:id="echoid-s16062" xml:space="preserve">P _Ag. </s>
  <s xml:id="echoid-s16063" xml:space="preserve">5. </s>
  <s xml:id="echoid-s16064" xml:space="preserve">Lin. </s>
  <s xml:id="echoid-s16065" xml:space="preserve">20_. </s>
  <s xml:id="echoid-s16066" xml:space="preserve">ad teſtatur, _lege_ teſtatam facit. </s>
  <s xml:id="echoid-s16067" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16068" xml:space="preserve">9. </s>
  <s xml:id="echoid-s16069" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16070" xml:space="preserve">3. </s>
  <s xml:id="echoid-s16071" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16072" xml:space="preserve">velocitatum. <lb/></s>
  <s xml:id="echoid-s16073" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16074" xml:space="preserve">14. </s>
  <s xml:id="echoid-s16075" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16076" xml:space="preserve">36. </s>
  <s xml:id="echoid-s16077" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16078" xml:space="preserve">plana. </s>
  <s xml:id="echoid-s16079" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16080" xml:space="preserve">17. </s>
  <s xml:id="echoid-s16081" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16082" xml:space="preserve">24. </s>
  <s xml:id="echoid-s16083" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16084" xml:space="preserve">prohibetur. </s>
  <s xml:id="echoid-s16085" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16086" xml:space="preserve">18. </s>
  <s xml:id="echoid-s16087" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16088" xml:space="preserve">32. </s>
  <s xml:id="echoid-s16089" xml:space="preserve">leg_ à puncto B. </s>
  <s xml:id="echoid-s16090" xml:space="preserve"><lb/>_p. </s>
  <s xml:id="echoid-s16091" xml:space="preserve">19. </s>
  <s xml:id="echoid-s16092" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16093" xml:space="preserve">4. </s>
  <s xml:id="echoid-s16094" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16095" xml:space="preserve">BD , GK . </s>
  <s xml:id="echoid-s16096" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16097" xml:space="preserve">22. </s>
  <s xml:id="echoid-s16098" xml:space="preserve">10. </s>
  <s xml:id="echoid-s16099" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16100" xml:space="preserve">VD multitudo cenſeri. </s>
  <s xml:id="echoid-s16101" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16102" xml:space="preserve">23. </s>
  <s xml:id="echoid-s16103" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16104" xml:space="preserve">7_. </s>
  <s xml:id="echoid-s16105" xml:space="preserve"><lb/>_leg_. </s>
  <s xml:id="echoid-s16106" xml:space="preserve">radius ad _p. </s>
  <s xml:id="echoid-s16107" xml:space="preserve">23. </s>
  <s xml:id="echoid-s16108" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16109" xml:space="preserve">10. </s>
  <s xml:id="echoid-s16110" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16111" xml:space="preserve">nec non, datis. </s>
  <s xml:id="echoid-s16112" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16113" xml:space="preserve">24. </s>
  <s xml:id="echoid-s16114" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16115" xml:space="preserve">2. </s>
  <s xml:id="echoid-s16116" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16117" xml:space="preserve">effectæ. </s>
  <s xml:id="echoid-s16118" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16119" xml:space="preserve">24. </s>
  <s xml:id="echoid-s16120" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16121" xml:space="preserve">24_. </s>
  <s xml:id="echoid-s16122" xml:space="preserve"><lb/>_leg_. </s>
  <s xml:id="echoid-s16123" xml:space="preserve">quidem ut punctum. </s>
  <s xml:id="echoid-s16124" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16125" xml:space="preserve">30. </s>
  <s xml:id="echoid-s16126" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16127" xml:space="preserve">18. </s>
  <s xml:id="echoid-s16128" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16129" xml:space="preserve">protracta. </s>
  <s xml:id="echoid-s16130" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16131" xml:space="preserve">32. </s>
  <s xml:id="echoid-s16132" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16133" xml:space="preserve">5, 6. </s>
  <s xml:id="echoid-s16134" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16135" xml:space="preserve">tangentes <lb/>(una - hujus) _p. </s>
  <s xml:id="echoid-s16136" xml:space="preserve">35. </s>
  <s xml:id="echoid-s16137" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16138" xml:space="preserve">5. </s>
  <s xml:id="echoid-s16139" xml:space="preserve">leg_ tangant. </s>
  <s xml:id="echoid-s16140" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16141" xml:space="preserve">35. </s>
  <s xml:id="echoid-s16142" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16143" xml:space="preserve">6. </s>
  <s xml:id="echoid-s16144" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16145" xml:space="preserve">MP . </s>
  <s xml:id="echoid-s16146" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16147" xml:space="preserve">35. </s>
  <s xml:id="echoid-s16148" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16149" xml:space="preserve">12. </s>
  <s xml:id="echoid-s16150" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16151" xml:space="preserve">TP . </s>
  <s xml:id="echoid-s16152" xml:space="preserve"><lb/>_p. </s>
  <s xml:id="echoid-s16153" xml:space="preserve">37. </s>
  <s xml:id="echoid-s16154" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16155" xml:space="preserve">2. </s>
  <s xml:id="echoid-s16156" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16157" xml:space="preserve">divisâ. </s>
  <s xml:id="echoid-s16158" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16159" xml:space="preserve">40. </s>
  <s xml:id="echoid-s16160" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16161" xml:space="preserve">4. </s>
  <s xml:id="echoid-s16162" xml:space="preserve">leg_ arcus NH major eſt ipsâ. </s>
  <s xml:id="echoid-s16163" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16164" xml:space="preserve">41. </s>
  <s xml:id="echoid-s16165" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16166" xml:space="preserve">32. </s>
  <s xml:id="echoid-s16167" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16168" xml:space="preserve">ver-<lb/>ſari. </s>
  <s xml:id="echoid-s16169" xml:space="preserve">_p 43. </s>
  <s xml:id="echoid-s16170" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16171" xml:space="preserve">15. </s>
  <s xml:id="echoid-s16172" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16173" xml:space="preserve">aliam HR . </s>
  <s xml:id="echoid-s16174" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16175" xml:space="preserve">47. </s>
  <s xml:id="echoid-s16176" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16177" xml:space="preserve">26._ </s>
  <s xml:id="echoid-s16178" xml:space="preserve">Fig 39, &amp; </s>
  <s xml:id="echoid-s16179" xml:space="preserve">40. </s>
  <s xml:id="echoid-s16180" xml:space="preserve">_pag. </s>
  <s xml:id="echoid-s16181" xml:space="preserve">49. </s>
  <s xml:id="echoid-s16182" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16183" xml:space="preserve">16. </s>
  <s xml:id="echoid-s16184" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16185" xml:space="preserve"><lb/>_@ f x y. </s>
  <s xml:id="echoid-s16186" xml:space="preserve">p. </s>
  <s xml:id="echoid-s16187" xml:space="preserve">52. </s>
  <s xml:id="echoid-s16188" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16189" xml:space="preserve">3. </s>
  <s xml:id="echoid-s16190" xml:space="preserve">dele_ Fig. </s>
  <s xml:id="echoid-s16191" xml:space="preserve">_51, 52. </s>
  <s xml:id="echoid-s16192" xml:space="preserve">p. </s>
  <s xml:id="echoid-s16193" xml:space="preserve">52. </s>
  <s xml:id="echoid-s16194" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16195" xml:space="preserve">6. </s>
  <s xml:id="echoid-s16196" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16197" xml:space="preserve">Fig. </s>
  <s xml:id="echoid-s16198" xml:space="preserve">_51, 52. </s>
  <s xml:id="echoid-s16199" xml:space="preserve">pag. </s>
  <s xml:id="echoid-s16200" xml:space="preserve">52 l. </s>
  <s xml:id="echoid-s16201" xml:space="preserve">24. </s>
  <s xml:id="echoid-s16202" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16203" xml:space="preserve"><lb/>Fig. </s>
  <s xml:id="echoid-s16204" xml:space="preserve">53. </s>
  <s xml:id="echoid-s16205" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16206" xml:space="preserve">55. </s>
  <s xml:id="echoid-s16207" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16208" xml:space="preserve">15. </s>
  <s xml:id="echoid-s16209" xml:space="preserve">dele_ ſe interſecantes in X. </s>
  <s xml:id="echoid-s16210" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16211" xml:space="preserve">57. </s>
  <s xml:id="echoid-s16212" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16213" xml:space="preserve">25. </s>
  <s xml:id="echoid-s16214" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16215" xml:space="preserve">δ P. </s>
  <s xml:id="echoid-s16216" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16217" xml:space="preserve">58. </s>
  <s xml:id="echoid-s16218" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16219" xml:space="preserve">19_. </s>
  <s xml:id="echoid-s16220" xml:space="preserve"><lb/>_leg._ </s>
  <s xml:id="echoid-s16221" xml:space="preserve">FB F ipſi KE K. </s>
  <s xml:id="echoid-s16222" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16223" xml:space="preserve">59. </s>
  <s xml:id="echoid-s16224" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16225" xml:space="preserve">2. </s>
  <s xml:id="echoid-s16226" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16227" xml:space="preserve">KE K. </s>
  <s xml:id="echoid-s16228" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16229" xml:space="preserve">61. </s>
  <s xml:id="echoid-s16230" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16231" xml:space="preserve">26. </s>
  <s xml:id="echoid-s16232" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16233" xml:space="preserve">punctum. </s>
  <s xml:id="echoid-s16234" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16235" xml:space="preserve">62. </s>
  <s xml:id="echoid-s16236" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16237" xml:space="preserve">27_. </s>
  <s xml:id="echoid-s16238" xml:space="preserve"><lb/>_leg._ </s>
  <s xml:id="echoid-s16239" xml:space="preserve">KO &amp;</s>
  <s xml:id="echoid-s16240" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s16241" xml:space="preserve">KA _p. </s>
  <s xml:id="echoid-s16242" xml:space="preserve">63. </s>
  <s xml:id="echoid-s16243" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16244" xml:space="preserve">16. </s>
  <s xml:id="echoid-s16245" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16246" xml:space="preserve">contactum. </s>
  <s xml:id="echoid-s16247" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16248" xml:space="preserve">64. </s>
  <s xml:id="echoid-s16249" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16250" xml:space="preserve">22. </s>
  <s xml:id="echoid-s16251" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16252" xml:space="preserve">Fig. </s>
  <s xml:id="echoid-s16253" xml:space="preserve">_80. </s>
  <s xml:id="echoid-s16254" xml:space="preserve">p. </s>
  <s xml:id="echoid-s16255" xml:space="preserve">65. </s>
  <s xml:id="echoid-s16256" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16257" xml:space="preserve">4_. </s>
  <s xml:id="echoid-s16258" xml:space="preserve"><lb/>_leg_. </s>
  <s xml:id="echoid-s16259" xml:space="preserve">ςενολεοχίαν. </s>
  <s xml:id="echoid-s16260" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16261" xml:space="preserve">67. </s>
  <s xml:id="echoid-s16262" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16263" xml:space="preserve">11. </s>
  <s xml:id="echoid-s16264" xml:space="preserve">leg_ tum alia. </s>
  <s xml:id="echoid-s16265" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16266" xml:space="preserve">67. </s>
  <s xml:id="echoid-s16267" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16268" xml:space="preserve">35. </s>
  <s xml:id="echoid-s16269" xml:space="preserve">leg_, QO q = Z q. </s>
  <s xml:id="echoid-s16270" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16271" xml:space="preserve">68_. </s>
  <s xml:id="echoid-s16272" xml:space="preserve"><lb/>_l. </s>
  <s xml:id="echoid-s16273" xml:space="preserve">7. </s>
  <s xml:id="echoid-s16274" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16275" xml:space="preserve">FQ . </s>
  <s xml:id="echoid-s16276" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16277" xml:space="preserve">70. </s>
  <s xml:id="echoid-s16278" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16279" xml:space="preserve">22. </s>
  <s xml:id="echoid-s16280" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16281" xml:space="preserve">Fig. </s>
  <s xml:id="echoid-s16282" xml:space="preserve">_95. </s>
  <s xml:id="echoid-s16283" xml:space="preserve">p. </s>
  <s xml:id="echoid-s16284" xml:space="preserve">76. </s>
  <s xml:id="echoid-s16285" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16286" xml:space="preserve">3. </s>
  <s xml:id="echoid-s16287" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16288" xml:space="preserve">HT (_a_) &amp;</s>
  <s xml:id="echoid-s16289" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s16290" xml:space="preserve">GA . </s>
  <s xml:id="echoid-s16291" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16292" xml:space="preserve">76_. </s>
  <s xml:id="echoid-s16293" xml:space="preserve"><lb/>_l. </s>
  <s xml:id="echoid-s16294" xml:space="preserve">11. </s>
  <s xml:id="echoid-s16295" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16296" xml:space="preserve">DF _p. </s>
  <s xml:id="echoid-s16297" xml:space="preserve">76. </s>
  <s xml:id="echoid-s16298" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16299" xml:space="preserve">18 leg_. </s>
  <s xml:id="echoid-s16300" xml:space="preserve">PK . </s>
  <s xml:id="echoid-s16301" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16302" xml:space="preserve">76. </s>
  <s xml:id="echoid-s16303" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16304" xml:space="preserve">20. </s>
  <s xml:id="echoid-s16305" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16306" xml:space="preserve">tanget recta RFK. </s>
  <s xml:id="echoid-s16307" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16308" xml:space="preserve">78. </s>
  <s xml:id="echoid-s16309" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16310" xml:space="preserve">24_. </s>
  <s xml:id="echoid-s16311" xml:space="preserve"><lb/>_leg_. </s>
  <s xml:id="echoid-s16312" xml:space="preserve">infra. </s>
  <s xml:id="echoid-s16313" xml:space="preserve">_p 79. </s>
  <s xml:id="echoid-s16314" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16315" xml:space="preserve">18. </s>
  <s xml:id="echoid-s16316" xml:space="preserve">dele_ Fig. </s>
  <s xml:id="echoid-s16317" xml:space="preserve">113. </s>
  <s xml:id="echoid-s16318" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16319" xml:space="preserve">79. </s>
  <s xml:id="echoid-s16320" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16321" xml:space="preserve">31. </s>
  <s xml:id="echoid-s16322" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16323" xml:space="preserve">Fig. </s>
  <s xml:id="echoid-s16324" xml:space="preserve">113. </s>
  <s xml:id="echoid-s16325" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16326" xml:space="preserve">86. </s>
  <s xml:id="echoid-s16327" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16328" xml:space="preserve">31_. </s>
  <s xml:id="echoid-s16329" xml:space="preserve"><lb/>_leg_. </s>
  <s xml:id="echoid-s16330" xml:space="preserve">√ VC Z φ = CG. </s>
  <s xml:id="echoid-s16331" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16332" xml:space="preserve">87 l. </s>
  <s xml:id="echoid-s16333" xml:space="preserve">14. </s>
  <s xml:id="echoid-s16334" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16335" xml:space="preserve">D Ψ<emph style="sub">3</emph> = √ {32/243}. </s>
  <s xml:id="echoid-s16336" xml:space="preserve">_pag. </s>
  <s xml:id="echoid-s16337" xml:space="preserve">91. </s>
  <s xml:id="echoid-s16338" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16339" xml:space="preserve">9. </s>
  <s xml:id="echoid-s16340" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16341" xml:space="preserve"><lb/>inrecta. </s>
  <s xml:id="echoid-s16342" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16343" xml:space="preserve">91. </s>
  <s xml:id="echoid-s16344" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16345" xml:space="preserve">23. </s>
  <s xml:id="echoid-s16346" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16347" xml:space="preserve">æquale rectangulo ex. </s>
  <s xml:id="echoid-s16348" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16349" xml:space="preserve">91. </s>
  <s xml:id="echoid-s16350" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16351" xml:space="preserve">24. </s>
  <s xml:id="echoid-s16352" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16353" xml:space="preserve">P, Q. </s>
  <s xml:id="echoid-s16354" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16355" xml:space="preserve">96_. </s>
  <s xml:id="echoid-s16356" xml:space="preserve"><lb/>_l. </s>
  <s xml:id="echoid-s16357" xml:space="preserve">15. </s>
  <s xml:id="echoid-s16358" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16359" xml:space="preserve">CA. </s>
  <s xml:id="echoid-s16360" xml:space="preserve">CD. </s>
  <s xml:id="echoid-s16361" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16362" xml:space="preserve">96. </s>
  <s xml:id="echoid-s16363" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16364" xml:space="preserve">22. </s>
  <s xml:id="echoid-s16365" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16366" xml:space="preserve">AD = {s/t}CA . </s>
  <s xml:id="echoid-s16367" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16368" xml:space="preserve">97. </s>
  <s xml:id="echoid-s16369" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16370" xml:space="preserve">2. </s>
  <s xml:id="echoid-s16371" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16372" xml:space="preserve">totam. </s>
  <s xml:id="echoid-s16373" xml:space="preserve"><lb/>_p. </s>
  <s xml:id="echoid-s16374" xml:space="preserve">102. </s>
  <s xml:id="echoid-s16375" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16376" xml:space="preserve">25. </s>
  <s xml:id="echoid-s16377" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16378" xml:space="preserve">OP ad OT . </s>
  <s xml:id="echoid-s16379" xml:space="preserve">_pag. </s>
  <s xml:id="echoid-s16380" xml:space="preserve">106. </s>
  <s xml:id="echoid-s16381" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16382" xml:space="preserve">10. </s>
  <s xml:id="echoid-s16383" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16384" xml:space="preserve">applicatis, _p. </s>
  <s xml:id="echoid-s16385" xml:space="preserve">106. </s>
  <s xml:id="echoid-s16386" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16387" xml:space="preserve">19. </s>
  <s xml:id="echoid-s16388" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16389" xml:space="preserve"><lb/>ſemi-axis. </s>
  <s xml:id="echoid-s16390" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16391" xml:space="preserve">112. </s>
  <s xml:id="echoid-s16392" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16393" xml:space="preserve">2. </s>
  <s xml:id="echoid-s16394" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16395" xml:space="preserve">applicatis. </s>
  <s xml:id="echoid-s16396" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16397" xml:space="preserve">114. </s>
  <s xml:id="echoid-s16398" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16399" xml:space="preserve">22. </s>
  <s xml:id="echoid-s16400" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16401" xml:space="preserve">{PLQO/2 Rad.</s>
  <s xml:id="echoid-s16402" xml:space="preserve">} _p. </s>
  <s xml:id="echoid-s16403" xml:space="preserve">114. </s>
  <s xml:id="echoid-s16404" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16405" xml:space="preserve">26_. </s>
  <s xml:id="echoid-s16406" xml:space="preserve"><lb/>_leg_. </s>
  <s xml:id="echoid-s16407" xml:space="preserve">propoſitum. </s>
  <s xml:id="echoid-s16408" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16409" xml:space="preserve">116. </s>
  <s xml:id="echoid-s16410" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16411" xml:space="preserve">5. </s>
  <s xml:id="echoid-s16412" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16413" xml:space="preserve">R. </s>
  <s xml:id="echoid-s16414" xml:space="preserve">S. </s>
  <s xml:id="echoid-s16415" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16416" xml:space="preserve">122. </s>
  <s xml:id="echoid-s16417" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16418" xml:space="preserve">22. </s>
  <s xml:id="echoid-s16419" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16420" xml:space="preserve">Fig. </s>
  <s xml:id="echoid-s16421" xml:space="preserve">_183. </s>
  <s xml:id="echoid-s16422" xml:space="preserve">p. </s>
  <s xml:id="echoid-s16423" xml:space="preserve">123. </s>
  <s xml:id="echoid-s16424" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16425" xml:space="preserve">1. </s>
  <s xml:id="echoid-s16426" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16427" xml:space="preserve"><lb/>Fig. </s>
  <s xml:id="echoid-s16428" xml:space="preserve">184 _p. </s>
  <s xml:id="echoid-s16429" xml:space="preserve">125. </s>
  <s xml:id="echoid-s16430" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16431" xml:space="preserve">4. </s>
  <s xml:id="echoid-s16432" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16433" xml:space="preserve">DM = DI . </s>
  <s xml:id="echoid-s16434" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16435" xml:space="preserve">128. </s>
  <s xml:id="echoid-s16436" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16437" xml:space="preserve">7. </s>
  <s xml:id="echoid-s16438" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16439" xml:space="preserve">Fig. </s>
  <s xml:id="echoid-s16440" xml:space="preserve">195. </s>
  <s xml:id="echoid-s16441" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16442" xml:space="preserve">128. </s>
  <s xml:id="echoid-s16443" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16444" xml:space="preserve">11:_ </s>
  <s xml:id="echoid-s16445" xml:space="preserve"><lb/>_dele_ Fig. </s>
  <s xml:id="echoid-s16446" xml:space="preserve">195. </s>
  <s xml:id="echoid-s16447" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16448" xml:space="preserve">128. </s>
  <s xml:id="echoid-s16449" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16450" xml:space="preserve">23_ {PM/√APM} _p. </s>
  <s xml:id="echoid-s16451" xml:space="preserve">129. </s>
  <s xml:id="echoid-s16452" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16453" xml:space="preserve">13_. </s>
  <s xml:id="echoid-s16454" xml:space="preserve">emerget undecima Lect. </s>
  <s xml:id="echoid-s16455" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16456" xml:space="preserve">136_. </s>
  <s xml:id="echoid-s16457" xml:space="preserve"><lb/>_l. </s>
  <s xml:id="echoid-s16458" xml:space="preserve">20. </s>
  <s xml:id="echoid-s16459" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16460" xml:space="preserve">hyperbolæ. </s>
  <s xml:id="echoid-s16461" xml:space="preserve">_pag. </s>
  <s xml:id="echoid-s16462" xml:space="preserve">139. </s>
  <s xml:id="echoid-s16463" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16464" xml:space="preserve">3. </s>
  <s xml:id="echoid-s16465" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16466" xml:space="preserve">nam. </s>
  <s xml:id="echoid-s16467" xml:space="preserve">_p. </s>
  <s xml:id="echoid-s16468" xml:space="preserve">140. </s>
  <s xml:id="echoid-s16469" xml:space="preserve">l. </s>
  <s xml:id="echoid-s16470" xml:space="preserve">1. </s>
  <s xml:id="echoid-s16471" xml:space="preserve">leg. </s>
  <s xml:id="echoid-s16472" xml:space="preserve">n &amp;</s>
  <s xml:id="echoid-s16473" xml:space="preserve">lt;</s>
  <s xml:id="echoid-s16474" xml:space="preserve">β_. </s>
  <s xml:id="echoid-s16475" xml:space="preserve"><lb/>_à p. </s>
  <s xml:id="echoid-s16476" xml:space="preserve">105_. </s>
  <s xml:id="echoid-s16477" xml:space="preserve">ad _p. </s>
  <s xml:id="echoid-s16478" xml:space="preserve">112. </s>
  <s xml:id="echoid-s16479" xml:space="preserve">l 1. </s>
  <s xml:id="echoid-s16480" xml:space="preserve">leg_. </s>
  <s xml:id="echoid-s16481" xml:space="preserve">Lect. </s>
  <s xml:id="echoid-s16482" xml:space="preserve">XII.</s>
  <s xml:id="echoid-s16483" xml:space="preserve">@</s>
</p>
<pb o="149" file="0327" n="342"/>
</div>
<div xml:id="echoid-div582" type="section" level="1" n="101">
<head xml:id="echoid-head105" xml:space="preserve">Addenda Lectionibus Geometricis.</head>
<p style="it">
  <s xml:id="echoid-s16484" xml:space="preserve">Vacuæ Pagellæ explendæ bæc adjici poſſunt: </s>
  <s xml:id="echoid-s16485" xml:space="preserve">υΠοραδικὰ vice, <lb/>animadverto potuiſſe ſecundo Appendiculæ tertiæ Lectio-<lb/>nis XII Problemati, pag. </s>
  <s xml:id="echoid-s16486" xml:space="preserve">122. </s>
  <s xml:id="echoid-s16487" xml:space="preserve">Corollaria quædam adponi <lb/>non injucunda, qualium adſcribam unum &amp; </s>
  <s xml:id="echoid-s16488" xml:space="preserve">alterum.</s>
  <s xml:id="echoid-s16489" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div583" type="section" level="1" n="102">
<head xml:id="echoid-head106" xml:space="preserve">_Probl_. I.</head>
<p>
  <s xml:id="echoid-s16490" xml:space="preserve">DE tur linea quæpiam AMB (cujus axis AD, baſis DB) <lb/>
<anchor type="note" xlink:label="note-0327-01a" xlink:href="note-0327-01"/>
curva AN E deſignetur talis, ut ductâ liberè rectà MN G <lb/>ad BD parallelâ, quæ ipſam AN E ſecet in N, ſit curva AN <lb/>æqualis ipſi GM .</s>
  <s xml:id="echoid-s16491" xml:space="preserve"/>
</p>
<div xml:id="echoid-div583" type="float" level="2" n="1">
<note position="right" xlink:label="note-0327-01" xlink:href="note-0327-01a" xml:space="preserve">Fig. 221.</note>
</div>
<p>
  <s xml:id="echoid-s16492" xml:space="preserve">Curva AN E talis ſit ut ſi MT curvam AMB, &amp; </s>
  <s xml:id="echoid-s16493" xml:space="preserve">NS cur-<lb/>vam ANE tangant, ſit SG. </s>
  <s xml:id="echoid-s16494" xml:space="preserve">GN :</s>
  <s xml:id="echoid-s16495" xml:space="preserve">: TG. </s>
  <s xml:id="echoid-s16496" xml:space="preserve">√ GM q - TG q, <lb/>ipſa ANE Propoſito faciet ſatis.</s>
  <s xml:id="echoid-s16497" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div585" type="section" level="1" n="103">
<head xml:id="echoid-head107" xml:space="preserve">_Probl_. II.</head>
<p>
  <s xml:id="echoid-s16498" xml:space="preserve">Iiſdem quoad cætera Suppoſitis, &amp; </s>
  <s xml:id="echoid-s16499" xml:space="preserve">conſtitutis; </s>
  <s xml:id="echoid-s16500" xml:space="preserve">curva ANE <lb/>jam talis eſſe debeat, ut curva AN ſemper æquetur interceptæ rectæ <lb/>NM.</s>
  <s xml:id="echoid-s16501" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16502" xml:space="preserve">Curva ANE jam talis ſit, ut ſit SG. </s>
  <s xml:id="echoid-s16503" xml:space="preserve">GN :</s>
  <s xml:id="echoid-s16504" xml:space="preserve">: 2 TG x GM. <lb/></s>
  <s xml:id="echoid-s16505" xml:space="preserve">GM q - TG q; </s>
  <s xml:id="echoid-s16506" xml:space="preserve">erit ANE curva quæ deſideratur.</s>
  <s xml:id="echoid-s16507" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div586" type="section" level="1" n="104">
<head xml:id="echoid-head108" xml:space="preserve">_Probl_. III.</head>
<p>
  <s xml:id="echoid-s16508" xml:space="preserve">Datur curva quæpiam DX X, cujus axis DA ; </s>
  <s xml:id="echoid-s16509" xml:space="preserve">reperiatur curva <lb/>
<anchor type="note" xlink:label="note-0327-02a" xlink:href="note-0327-02"/>
AM B proprietate talis, ut ſi liberè ducatur recta GX M ad ipſam <lb/>AD perpendicularis, ponaturque SM T curvam AM tangere, ſit <lb/>MS æqualis ipſi GX .</s>
  <s xml:id="echoid-s16510" xml:space="preserve"/>
</p>
<div xml:id="echoid-div586" type="float" level="2" n="1">
<note position="right" xlink:label="note-0327-02" xlink:href="note-0327-02a" xml:space="preserve">Fig. 222.</note>
</div>
<p>
  <s xml:id="echoid-s16511" xml:space="preserve">Liquetrationem TG ad TM (hoc eſt rationem GD ad MS, vel <lb/>GX ) dari; </s>
  <s xml:id="echoid-s16512" xml:space="preserve">adeoque rationem TG ad GM quoque dari.</s>
  <s xml:id="echoid-s16513" xml:space="preserve"/>
</p>
<pb o="150" file="0328" n="343"/>
</div>
<div xml:id="echoid-div588" type="section" level="1" n="105">
<head xml:id="echoid-head109" style="it" xml:space="preserve">Addenda Lectionibus Geometricis.</head>
<p>
  <s xml:id="echoid-s16514" xml:space="preserve">Inſervit hoc ſuperficiebus deſignandis, quarum in promptu ſit di-<lb/>menſio, etenim (ductâ ME ad AD parallelâ) Superficies Solidi <lb/>ex plani BM E circa axem DB rotatu progeniti adæquat {Periph/Rad} <lb/>x GD X; </s>
  <s xml:id="echoid-s16515" xml:space="preserve">ut habetur in 11<emph style="sub">a</emph> Lectionis XII.</s>
  <s xml:id="echoid-s16516" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s16517" xml:space="preserve">In Lect. </s>
  <s xml:id="echoid-s16518" xml:space="preserve">XI. </s>
  <s xml:id="echoid-s16519" xml:space="preserve">appendice, numero XXXIII. </s>
  <s xml:id="echoid-s16520" xml:space="preserve">de Cycloide profer-<lb/>tur Tbeorema quoddam, id quod ex bujuſmodi generaliori <lb/>Tbeoremate deduci potuiſſet.</s>
  <s xml:id="echoid-s16521" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16522" xml:space="preserve">SI t AM B curva quælibet, cujus Axis AD , baſis DB , ſit item <lb/>
<anchor type="note" xlink:label="note-0328-01a" xlink:href="note-0328-01"/>
curva AN E talis, ut ſi arbitrariè ducatur PM N ad DB E pa-<lb/>rallela, poſitoque rectam TN curvam AN E tangere, ſit TN parallela <lb/>ſubtenſæ AM ; </s>
  <s xml:id="echoid-s16523" xml:space="preserve">completo Rectangulo AD EG erit Spatium trili-<lb/>neum AE G æquale Segmento AD B.</s>
  <s xml:id="echoid-s16524" xml:space="preserve"/>
</p>
<div xml:id="echoid-div588" type="float" level="2" n="1">
<note position="left" xlink:label="note-0328-01" xlink:href="note-0328-01a" xml:space="preserve">Fig. 223.</note>
</div>
<p>
  <s xml:id="echoid-s16525" xml:space="preserve">Huic ſuppar Theorema tale eſt: </s>
  <s xml:id="echoid-s16526" xml:space="preserve">liſdem poſitis, ſi tam Segmentum <lb/>AD B, quam Spatium AE G circa Axem AG convertantur; </s>
  <s xml:id="echoid-s16527" xml:space="preserve">erit <lb/>productum è Segmento AD BS olidum producti ex AE G duplum.</s>
  <s xml:id="echoid-s16528" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16529" xml:space="preserve">E tangentium porrò contemplatione ſuborta eſt methodus, per <lb/>quam expediſſimè plurima circa maximas quantitates Theoremata <lb/>deducuntur; </s>
  <s xml:id="echoid-s16530" xml:space="preserve">quæ certè ſi tempeſtivè ſe objeciſſent, digna cenſuiſſem <lb/>quæ Lectionibus inſererentur, ex iis indigitabo nonnulla.</s>
  <s xml:id="echoid-s16531" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16532" xml:space="preserve">Sit curva quæpiam AL B, cujus Axis AD , baſis DB ; </s>
  <s xml:id="echoid-s16533" xml:space="preserve">&amp; </s>
  <s xml:id="echoid-s16534" xml:space="preserve">huic <lb/>
<anchor type="note" xlink:label="note-0328-02a" xlink:href="note-0328-02"/>
parallelæ LG , λ γ; </s>
  <s xml:id="echoid-s16535" xml:space="preserve">item LT curvam tangat.</s>
  <s xml:id="echoid-s16536" xml:space="preserve"/>
</p>
<div xml:id="echoid-div589" type="float" level="2" n="2">
<note position="right" xlink:label="note-0328-02" xlink:href="note-0328-02a" xml:space="preserve">Fig. 224.</note>
</div>
</div>
<div xml:id="echoid-div591" type="section" level="1" n="106">
<head xml:id="echoid-head110" xml:space="preserve">_Theor_. I.</head>
<p>
  <s xml:id="echoid-s16537" xml:space="preserve">Sit _m_ numerus quicunque, poteſtates exponens; </s>
  <s xml:id="echoid-s16538" xml:space="preserve">ſi ponatur <lb/>DG {_m_ - 1/} x TG = GL {_m_/}, erit DG {_m_/} + GL {_m_/} maximum, ſeu <lb/>majus quam D γ {_m_/} + γ λ {_m_/}.</s>
  <s xml:id="echoid-s16539" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div592" type="section" level="1" n="107">
<head xml:id="echoid-head111" xml:space="preserve">_Theor_. II.</head>
<p>
  <s xml:id="echoid-s16540" xml:space="preserve">Itidem ſumpto numero _m_, ſi ponatur BL {_m_ - 1/} x TL = GL {_m_/}; <lb/></s>
  <s xml:id="echoid-s16541" xml:space="preserve">erit GL {_m_/} + BL {_m_/} maximum ſeu majus quam γ λ {_m_/} + B λ {_m_/}.</s>
  <s xml:id="echoid-s16542" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div593" type="section" level="1" n="108">
<head xml:id="echoid-head112" xml:space="preserve">_Theor_. III.</head>
<p>
  <s xml:id="echoid-s16543" xml:space="preserve">Sint numeri quilibet _m_, _n_; </s>
  <s xml:id="echoid-s16544" xml:space="preserve">ſi ponatur _m_ x TG = _n_ x DG , erit <lb/>DG {_m_/} x GL {_n_/} maximum, ſeu majus quam D γ {_m_/} x γ λ {_n_/}.</s>
  <s xml:id="echoid-s16545" xml:space="preserve"/>
</p>
<pb o="151" file="0329" n="344" rhead=""/>
</div>
<div xml:id="echoid-div594" type="section" level="1" n="109">
<head xml:id="echoid-head113" xml:space="preserve">_Theor_. IV.</head>
<p>
  <s xml:id="echoid-s16546" xml:space="preserve">Quod ſi ponatur _m_ x TL = _n_ x arc BL , erit GL {_n_/} x BL {_m_/} <lb/>maximum, ſeu majus quàm γ λ {_n_/} x B λ {_m_/}.</s>
  <s xml:id="echoid-s16547" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div595" type="section" level="1" n="110">
<head xml:id="echoid-head114" xml:space="preserve">_Theor_. V.</head>
<p>
  <s xml:id="echoid-s16548" xml:space="preserve">Si fuerit TG x GL = DG LB, erit DG LB x GL maxi-<lb/>mum, ſeu majus quàm D γ λ B x γ λ.</s>
  <s xml:id="echoid-s16549" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div596" type="section" level="1" n="111">
<head xml:id="echoid-head115" xml:space="preserve">_Theor_. VI.</head>
<p>
  <s xml:id="echoid-s16550" xml:space="preserve">Sin TG x GL = 2 DG LB, erit GL x √ DG LB maxi-<lb/>mum, ſeu majus quàm γ λ x √ D γ λ B.</s>
  <s xml:id="echoid-s16551" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16552" xml:space="preserve">Haud difficili negotio, cum hæc demonſtrantur, tum ejuſmodi <lb/>complura deprehenduntur.</s>
  <s xml:id="echoid-s16553" xml:space="preserve"/>
</p>
<p style="it">
  <s xml:id="echoid-s16554" xml:space="preserve">Ad illa verò ſuccinctius comprobanda deſervire poſſunt bujuſmodi <lb/>Tbeoremata.</s>
  <s xml:id="echoid-s16555" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16556" xml:space="preserve">Sint duæ curvæ AG B, DH C quarum communis axis AD , <lb/>
<anchor type="note" xlink:label="note-0329-01a" xlink:href="note-0329-01"/>
ſed ordinatæ inverſo ſitu increſcant ab A ad DB , decreſcant à D ad <lb/>AC ; </s>
  <s xml:id="echoid-s16557" xml:space="preserve">ad ordinatæ verò communis GE H terminos, recta GS cur-<lb/>vam AG B, &amp; </s>
  <s xml:id="echoid-s16558" xml:space="preserve">recta HT curvam DH C contingant.</s>
  <s xml:id="echoid-s16559" xml:space="preserve"/>
</p>
<div xml:id="echoid-div596" type="float" level="2" n="1">
<note position="right" xlink:label="note-0329-01" xlink:href="note-0329-01a" xml:space="preserve">Fig. 225.</note>
</div>
<p>
  <s xml:id="echoid-s16560" xml:space="preserve">I. </s>
  <s xml:id="echoid-s16561" xml:space="preserve">Si recta HT rectæ GS parallela ſit, erit GE H maxima or-<lb/>dinatarum in continuum jacentium ſumma.</s>
  <s xml:id="echoid-s16562" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16563" xml:space="preserve">Nam utcunque ducta OK FL P ad GE H parallela (quæ Li-<lb/>neas ſecet ut cernis) erit GH = QP &amp;</s>
  <s xml:id="echoid-s16564" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s16565" xml:space="preserve">KL .</s>
  <s xml:id="echoid-s16566" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16567" xml:space="preserve">Not. </s>
  <s xml:id="echoid-s16568" xml:space="preserve">Verum hoc, ſi curvarum partes concavæ axi obverſæ jaceant, <lb/>aliàs GE H erit minima.</s>
  <s xml:id="echoid-s16569" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16570" xml:space="preserve">II. </s>
  <s xml:id="echoid-s16571" xml:space="preserve">Si ES = ET, erit rectangulum ex EG , EH maximum: <lb/></s>
  <s xml:id="echoid-s16572" xml:space="preserve">Nam ob SE. </s>
  <s xml:id="echoid-s16573" xml:space="preserve">SF :</s>
  <s xml:id="echoid-s16574" xml:space="preserve">: EG. </s>
  <s xml:id="echoid-s16575" xml:space="preserve">FO , &amp; </s>
  <s xml:id="echoid-s16576" xml:space="preserve">TE. </s>
  <s xml:id="echoid-s16577" xml:space="preserve">TF :</s>
  <s xml:id="echoid-s16578" xml:space="preserve">: EH. </s>
  <s xml:id="echoid-s16579" xml:space="preserve">FP ; </s>
  <s xml:id="echoid-s16580" xml:space="preserve">erit <lb/>SE x TE. </s>
  <s xml:id="echoid-s16581" xml:space="preserve">SF x TF :</s>
  <s xml:id="echoid-s16582" xml:space="preserve">: EG x EH. </s>
  <s xml:id="echoid-s16583" xml:space="preserve">FO x FP, itaque cum ſit <lb/>SE x TE &amp;</s>
  <s xml:id="echoid-s16584" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s16585" xml:space="preserve">SF x TF, erit EG x EH &amp;</s>
  <s xml:id="echoid-s16586" xml:space="preserve">gt; </s>
  <s xml:id="echoid-s16587" xml:space="preserve">FO x FP.</s>
  <s xml:id="echoid-s16588" xml:space="preserve"/>
</p>
</div>
<div xml:id="echoid-div598" type="section" level="1" n="112">
<head xml:id="echoid-head116" xml:space="preserve">FINIS.</head>
<pb file="0330" n="345"/>
<pb file="0331" n="346"/>
<p>
  <s xml:id="echoid-s16589" xml:space="preserve">UBi (_pag. </s>
  <s xml:id="echoid-s16590" xml:space="preserve">100_) de Centro gravitatis parabolæ &amp; </s>
  <s xml:id="echoid-s16591" xml:space="preserve">para-<lb/>boliformis verba fiunt, intelligantur non curvæ lineæ, <lb/>ſed iis comprehenſa ſpatia, de quibus apparet iſthic agi.</s>
  <s xml:id="echoid-s16592" xml:space="preserve"/>
</p>
<p>
  <s xml:id="echoid-s16593" xml:space="preserve">Sicubi ponitur {δ/π}, nec adponitur εκθεσις ulla, deſignan-<lb/>tur termini rationem exprimentes, quam habet circuli di-<lb/>ameter ad ejus circumferentiam.</s>
  <s xml:id="echoid-s16594" xml:space="preserve"/>
</p>
<pb file="0332" n="347"/>
<pb file="0333" n="348"/>
<pb o="1" file="0333a" n="349"/>
  <figure>
    <image file="0333a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0333a-01"/>
  </figure>
<pb file="0334" n="350"/>
<pb file="0335" n="351"/>
<pb o="2" file="0335a" n="352"/>
  <figure>
    <image file="0335a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0335a-01"/>
  </figure>
<pb file="0336" n="353"/>
<pb file="0337" n="354"/>
<pb o="3" file="0337a" n="355"/>
  <figure>
    <image file="0337a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0337a-01"/>
  </figure>
<pb file="0338" n="356"/>
<pb file="0339" n="357"/>
<pb o="4" file="0339a" n="358"/>
  <figure>
    <image file="0339a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0339a-01"/>
  </figure>
<pb file="0340" n="359"/>
<pb file="0341" n="360"/>
<pb o="5" file="0341a" n="361"/>
  <figure>
    <image file="0341a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0341a-01"/>
  </figure>
<pb file="0342" n="362"/>
<pb file="0343" n="363"/>
<pb o="6" file="0343a" n="364"/>
  <figure>
    <image file="0343a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0343a-01"/>
  </figure>
<pb file="0344" n="365"/>
<pb file="0345" n="366"/>
<pb o="7" file="0345a" n="367"/>
  <figure>
    <image file="0345a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0345a-01"/>
  </figure>
<pb file="0346" n="368"/>
<pb file="0347" n="369"/>
<pb o="8" file="0347a" n="370"/>
  <figure>
    <image file="0347a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0347a-01"/>
  </figure>
<pb file="0348" n="371"/>
<pb file="0349" n="372"/>
<pb o="9" file="0349a" n="373"/>
  <figure>
    <image file="0349a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0349a-01"/>
  </figure>
<pb file="0350" n="374"/>
<pb file="0351" n="375"/>
<pb o="10" file="0351a" n="376"/>
  <figure>
    <image file="0351a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0351a-01"/>
  </figure>
<pb file="0352" n="377"/>
<pb file="0353" n="378"/>
<pb o="11" file="0353a" n="379"/>
  <figure>
    <image file="0353a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0353a-01"/>
  </figure>
<pb file="0354" n="380"/>
<pb file="0355" n="381"/>
<pb o="12" file="0355a" n="382"/>
  <figure>
    <image file="0355a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0355a-01"/>
  </figure>
<pb file="0356" n="383"/>
<pb file="0357" n="384"/>
<pb o="13" file="0357a" n="385"/>
  <figure>
    <image file="0357a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/U19ERSE3/figures/0357a-01"/>
  </figure>
<pb file="0358" n="386"/>
<pb file="0359" n="387"/>
<pb file="0360" n="388"/>
<pb file="0361" n="389"/>
<handwritten/>
<pb file="0362" n="390"/>
<pb file="0363" n="391"/>
<handwritten/>
<pb file="0364" n="392"/>
  </div></text>
</echo>
author	Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date	Wed, 29 Nov 2017 16:55:37 +0100
parents	22d6a63640c6
children